Minimal dynamical systems on the product

of the Cantor set and the circle

Huaxin Lin and Hiroki Matui

Abstract

We prove that a crossed product algebra arising from a minimal dynamical system on theproduct of the Cantor set and the circle has real rank zero if and only if the set of invariantmeasures of the system come from the associated Cantor minimal system. In the case thatcocycles take values in the rotation group, it is also shown that this condition implies tracialrank zero, and in particular, the crossed product algebra is isomorphic to a unital simpleAT-algebra of real rank zero. Under the same assumption, we show that two systems areapproximately K-conjugate if and only if there exists a sequence of isomorphisms betweentwo associated crossed products which approximately maps C(X × T) onto C(X × T).

1 Introduction

A celebrated theorem of Giordano, Putnam and Skau [GPS] gave a dynamical characterizationof isomorphism of the crossed product C∗-algebras arising from minimal dynamical systemson the Cantor set. The C∗-algebra theoretic aspect of this result is indebted to the fact thatthe algebras are unital simple AT-algebras with real rank zero. From the work of Q. Lin andN. C. Phillips [LP2] and the classification of simple C∗-algebras of tracial rank zero (see [L4]and [L5]), crossed product C∗-algebras arising from minimal diffeomorphisms on a manifold areisomorphic if they have the same Elliott invariants in the case that they have real rank zero (seealso [LP1]). However, there are no longer any dynamical characterizations of isomorphism ofthese algebras. The notion of approximate conjugacy was introduced in [LM] where it was alsosuggested that certain approximate version of conjugacy may be a right equivalence relation toensure isomorphism of crossed product C∗-algebras. Indeed, a complete description was givenfor Cantor minimal systems. In the present paper, we consider minimal dynamical systems onthe product of the Cantor set X and the circle T, and analyze the associated crossed productC∗-algebras and approximate conjugacy for those systems.

Since the Cantor set X is totally disconnected and the circle T is connected, every minimaldynamical system on X ×T can be viewed as a skew product extension of a minimal dynamicalsystem on X. This observation enables us to find a nice “large” subalgebra of the crossed productC∗-algebra in a similar fashion to the Cantor case. By applying results of [Ph2], it will beshown that the algebra has stable rank one and satisfies Blackadar’s fundamental comparabilityproperty. Moreover, we also prove that the algebra has real rank zero if and only if everyinvariant measure on X uniquely extends to an invariant measure on X × T. Such a system issaid to be rigid. As a special case we consider cocycles taking their values in the rotation groupand show that the algebra has tracial rank zero if and only if the system is rigid (see Definition3.1). By an easy computation of K-groups and the classification theorem of [L5], we concludethat the associated crossed product C∗-algebras are actually unital simple AT-algebras of realrank zero.

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A natural definition of approximate conjugacy for two dynamical systems (X,α) and (Y, β)is the following: there exists a sequence of homeomorphisms σn : X → Y such that

limn→∞

∥f ◦ σnασ−1n − f ◦ β∥ = 0

for every f ∈ C(Y ). If such a sequence exists, we can construct an asymptotic homomorphismbetween the associated crossed product C∗-algebras. In [LM], however, it was shown that thissimple relation is too weak for Cantor minimal systems. This happens because there are noconsistency in the sequence {σn}. Moreover, we proved in [M3] that a similar result holds forminimal dynamical systems on the product of the Cantor set X and the circle T. To obtaina stronger relation, one should impose additional conditions on the conjugating maps σn. Werequire that σn eventually induces the same map on K-theory. Approximate K-conjugacy wasintroduced in [LM] for Cantor minimal systems, and it was shown that two minimal systemsare approximately K-conjugate if and only if the associated crossed products are isomorphic.In the present paper, we will show that two minimal systems on X ×T associated with cocycleswith values in the rotation group are approximately K-conjugate if and only if there is an orderand unit preserving isomorphism between the K-theory of two associated crossed productswhich preserves the images of K-theory of C(X × T). In fact, in the case that both systemsare rigid, when two systems are approximately K-conjugate, the associated crossed products areisomorphic. But more is true. They are C∗-strongly approximately flip conjugate (see Definition7.4 below). The only difference from the case of Cantor minimal systems is that we have tocontrol a special projection which does not come from any clopen subsets of X. We call thatprojection the generalized Rieffel projection. Difference of two generalized Rieffel projectionswill be written by a Bott element associated with two almost commuting unitaries. A techniquedeveloped in [M3, Section 4] will be used more carefully in order to fix the position of thatprojection in the K0-group.

Non-orientation preserving cases are also discussed. By using a Z2-extension, we can untwista non-orientation preserving system and obtain an orientation preserving system. The associatedcrossed product C∗-algebra turns out to be isomorphic to the fixed point algebra of a Z2-actionon the crossed product associated with the orientation preserving system. Relation of theirK-groups is studied.

This paper is organized as follows. In Section 2, we collect notations and terminologiesrelevant to this paper and establish a few elementary facts. In Section 3, we investigate realrank, stable rank and comparability of projections of the crossed product C∗-algebra. Section 4is devoted to the case that cocycles take values in the rotation group. In Section 5, when cocyclestake values in the rotation group, we will prove that rigidity implies tracial rank zero. In Section6, the generalized Rieffel projection is defined. In Section 7, we show that isomorphism of K-groups implies approximate K-conjugacy when the systems arise from cocycles with values inthe rotation group. In Section 8, we deal with non-orientation preserving cases. In Section 9,we treat examples of various cocycles.

Acknowledgement The first named author would like to acknowledge the support from aNSF grant. He would also like to thank the second author for his effort to make this projectpossible. The second named author was supported by Grant-in-Aid for Young Scientists (B) ofJapan Society for the Promotion of Science. He is grateful to Yoshimichi Ueda and HiroyukiOsaka for helpful advises.

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2 Preliminaries

Let A be a unital C∗-algebra. We denote the union of Mk(A) for k = 1, 2, . . . by M∞(A).The K0-group of A is equipped with the order unit [1A] and the positive cone K0(A)+. Ahomomorphism s : K0(A) → R is called a state if s carries the order unit to one and the positivecone to nonnegative real numbers. We write the set of all states by S(K0(A)) and call it the statespace. Let T (A) denote the set of all tracial states on A. Endowed with the weak-∗ topology,T (A) is a compact convex set. The space of real valued affine continuous functions on T (A) iswritten by Aff(T (A)). We denote by D the natural homomorphism from K0(A) to Aff(T (A)).Namely, D([p])(τ) is equal to τ(p), where p is a projection of M∞(A) and τ is a tracial state onA. We say that the algebra A satisfies Blackadar’s second fundamental comparability questionwhen the order on projections of M∞(A) is determined by traces, that is, if p, q ∈ M∞(A) areprojections and τ(p) < τ(q) for all τ ∈ T (A), then p is Murray-von Neumann equivalent to asubprojection of q.

Let X be a compact metrizable space. Equip Homeo(X) with the topology of pointwiseconvergence in norm on C(X). Thus a sequence {αn}n∈N in Homeo(X) converges to α, if

limn→∞

supx∈X

|f(α−1n (x)) − f(α−1(x))| = 0

for every complex valued continuous function f ∈ C(X). This is equivalent to say that

supx∈X

d(αn(x), α(x))

tends to zero as n → ∞, where d(·, ·) is a metric which induces the topology of X. When Xis the Cantor set, this is also equivalent to say that, for any clopen subset U ⊂ X, there existsN ∈ N such that αn(U) = α(U) for all n ≥ N .

When α : X → X is a homeomorphism on a compact metrizable space X, we denote thecrossed product C∗-algebra arising from the dynamical system (X, α) by C∗(X,α). We use thenotation ufu∗ = f ◦ α−1, where f is a function on X and u is the implementing unitary. IfX is infinite and α is minimal, then C∗(X, α) is a simple C∗-algebra. We regard C(X) as asubalgebra of C∗(X,α). But when we need to emphasize the embedding, it will be denoted byjα : C(X) → C∗(X, α). Let Mα denote the set of α-invariant probability measures on X. If α isfree, then there exists a canonical bijection between Mα and the tracial state space T (C∗(X,α)).We may identify these spaces. Let (Y, β) be another dynamical system. A continuous surjectionF : X → Y is called a factor map if βF = Fα. It is well-known that F yields an affine continuoussurjection F∗ : Mα → Mβ by F∗(µ)(E) = µ(F−1(E)) for µ ∈ Mα and E ⊂ Y . We also remarkthat F induces a natural embedding of C∗(Y, β) into C∗(X, α).

When α is a minimal homeomorphism on the Cantor set X, we call (X, α) a Cantor minimalsystem. We briefly review results of [GPS]. The crossed product C∗-algebra C∗(X,α) is a unitalsimple AT algebra of real rank zero, and so it can be classified by its K-groups. The K1-groupof C∗(X, α) is isomorphic to Z, and K0(C∗(X, α)) is unital order isomorphic to

K0(X, α) = C(X, Z)/{f − f ◦ α−1 : f ∈ C(X, Z)}

equipped with the positive cone

K0(X, α)+ = {[f ] : f ∈ C(X, Z), f ≥ 0}

and the order unit [1X ], where [f ] means its equivalence class. We sometimes write [f ]α tospecify α. Throughout this paper, K0(C∗(X,α)) will be identified with K0(X, α). By [GPS,Theorem 2.1], K0(X, α) is a complete invariant for strong orbit equivalence of Cantor minimal

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systems. The idea of Kakutani-Rohlin partitions will be used repeatedly. We refer the readerto [HPS, Theorem 4.2] or [M3, Section 2] for details.

We identify the circle with T = R/Z and write the distance from t ∈ T to zero in T by|t|. Let Rt denote the translation on T = R/Z by t ∈ T. Then {Rt : t ∈ T} forms an abeliansubgroup of Homeo(T). We call it the rotation group. The set of isometric homeomorphisms onT is written by Isom(T). Thus,

Isom(T) = {Rt : t ∈ T} ∪ {Rtλ : t ∈ T},

where λ ∈ Homeo(T) is defined by λ(t) = −t. The finite cyclic group of order m is denoted byZm

∼= Z/mZ and may be identified with {0, 1, . . . ,m − 1}.Define o : Homeo(T) → Z2 by

o(ϕ) =

{0 ϕ is orientation preserving1 ϕ is orientation reversing.

Then the map o(·) is a homomorphism. Let Homeo+(T) denote the set of orientation preservinghomeomorphisms. Note that Homeo+(T) ∩ Isom(T) consists of rotations.

In the rest of this section, we establish notation and some elementary facts concerningdynamical systems on the product of the Cantor set X and the circle T.

Lemma 2.1. Let γ be a homeomorphism on X × T. Then, there exist α ∈ Homeo(X) and acontinuous map ϕ : X → Homeo(T) such that γ(x, t) = (α(x), ϕx(t)) for all (x, t) ∈ X × T.

Proof. This is obvious because the connected component including (x, t) is {x}×T and it mustbe carried to a connected component by the homeomorphism γ.

We denote the homeomorphism of the form in the lemma above by α × ϕ for short. Whenϕx = Rξ(x) with a continuous function ξ : X → T, we write α × Rξ. Let F : X × T → X be theprojection onto the first coordinate. Then we have F ◦ (α × ϕ) = α ◦ F . Thus, F is a factormap from (X × T, α × ϕ) to (X, α), and so F induces an affine continuous map from the set ofinvariant measures of (X × T, α × ϕ) to that of (X,α). Note that if α × ϕ is minimal then α isalso minimal.

We say that ϕ,ψ : X → Homeo(T) are cohomologous, when there exists a continuous mapω : X → Homeo(T) such that ψxωx = ωα(x)ϕx for all x ∈ X. If ϕ and ψ are cohomologous, itcan be easily verified that α × ϕ and α × ψ are conjugate.

Let o(ϕ) be the composition of ϕ : X → Homeo(T) and o : Homeo(T) → Z2, that is,o(ϕ)(x) = o(ϕx). Under the identification of

C(X, Z2)/{f − fα−1 : f ∈ C(X, Z2)}

with K0(X, α)/2K0(X, α) (see [M1, Lemma 3.5]), an element of K0(X, α)/2K0(X, α) is obtainedfrom o(ϕ). We write it by [o(ϕ)] or [o(ϕ)]α.

Definition 2.2. Let (X, α) be a Cantor minimal system and ϕ : X → Homeo(T) be a continuousmap. We say that α×ϕ or ϕ is orientation preserving, if [o(ϕ)] is zero in K0(X, α)/2K0(X, α).

Notice that the concept of ‘orientation reversing’ does not make sense in this situation andthat (α × ϕ)2 may not be orientation preserving.

Lemma 2.3 ([M3, Lemma 4.3]). Let (X, α) be a Cantor minimal system. If α × ϕ is anorientation preserving homeomorphism on X × T, then there exists a continuous map ψ : X →Homeo+(T) such that ϕ is cohomologous to ψ.

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Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuous map.We would like to compute the K-groups of C∗(X ×T, α×ϕ). Let us begin with the orientationpreserving case. When [o(ϕ)] is zero in K0(X, α)/2K0(X,α), by Lemma 2.3, we may assumethat o(ϕ)(x) is zero for all x ∈ X. It is evident that α × ϕ induces the action

α∗ : f 7→ f ◦ α−1

on K0(C(X ×T)) ∼= C(X, Z), and that the kernel of id−α∗ is spanned by [1X ] and the cokernelof id−α∗ is K0(X, α). Since o(ϕ)(x) = 0 for all x ∈ X, the induced action on K1(C(X × T)) ∼=C(X, Z) is α∗, too. Consequently we have the following.

Lemma 2.4. Let (X, α) be a Cantor minimal system and let α×ϕ be an orientation preservinghomeomorphism on X × T. Then both K0(C∗(X × T, α × ϕ)) and K1(C∗(X × T, α × ϕ)) areisomorphic to Z ⊕ K0(X, α).

Of course the embedding C∗(X, α) ⊂ C∗(X × T, α× ϕ) induces the embedding of K0(X, α)into K0(C∗(X ×T, α×ϕ)) preserving the unit. When α×ϕ is minimal, it can be easily verifiedthat this is really an order embedding. The order structure of the whole K0-group will beapparent in later sections.

Next, let us consider the non-orientation preserving case. Suppose that [o(ϕ)] is not zeroin K0(X,α)/2K0(X, α). Clearly α × ϕ induces the same action on K0(C(X × T)) ∼= C(X, Z)as the orientation preserving case. But, on the K1-group, the induced action is different. It isgiven by

α∗ϕ(f)(x) = (−1)o(ϕ)(α−1(x))f(α−1(x))

for f ∈ C(X, Z). We need to know the kernel and the cokernel of id−α∗ϕ. Suppose that f ∈

C(X, Z) belongs to Ker(id−α∗ϕ) and f = 0. Since |f(x)| = |α∗

ϕ(f)(x)| = |α∗(f)(x)| for all x ∈ X,the minimality of α implies that |f(x)| is a constant function. Define c ∈ C(X, Z2) by f(x) =(−1)c(x)|f(x)|. Then f(x) = (−1)o(ϕ)(α−1(x))f(α−1(x)) yields c(x) = o(ϕ)(α−1(x)) + c(α−1(x)),which contradicts [o(ϕ)] = 0. Hence id−α∗

ϕ is injective. It follows that K0(C∗(X ×T, α×ϕ)) ∼=K0(X, α).

Let (X ×Z2, α× o(ϕ)) be the skew product extension associated with the Z2-valued cocycleo(ϕ). As [o(ϕ)] = 0, (X ×Z2, α× o(ϕ)) is a Cantor minimal system (see [M1, Lemma 3.6]). Bydefinition, K0(X×Z2, α×o(ϕ)) is isomorphic to the cokernel of id−(α×o(ϕ))∗ on C(X×Z2, Z).Let π be the projection from X × Z2 onto the first coordinate. Then π is a factor map from(X×Z2, α×o(ϕ)) to (X, α). It is well-known that [f ] 7→ [f ◦π] is an order embedding between theK0-groups, and so we will regard K0(X, α) as a subgroup of K0(X×Z2, α×o(ϕ)). It is convenientto introduce a monomorphism δ from C(X, Z) to C(X×Z2, Z) defined by δ(f)(x, k) = (−1)kf(x).Then we can check that

δ ◦ α∗ϕ = (α × o(ϕ))∗ ◦ δ.

Define γ ∈ Homeo(X ×Z2) by γ(x, k) = (x, k + 1). Note that γ commutes with α× o(ϕ). SinceIm δ = Im(id−γ∗), we have

Coker(id−α∗ϕ) ∼= Im δ/ Im δ ◦ (id−α∗

ϕ)

= Im(id−γ∗)/ Im((id−(α × o(ϕ))∗) ◦ (id−γ∗))∼= C(X × Z2, Z)/(Ker(id−γ∗) + Im(id−(α × o(ϕ))∗))∼= K0(X × Z2, α × o(ϕ))/K0(X,α).

We can summarize the conclusion just obtained as follows.

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Lemma 2.5. Let (X, α) be a Cantor minimal system and let α × ϕ be a non-orientation pre-serving homeomorphism on X × T. Then we have

K0(C∗(X × T, α × ϕ)) ∼= K0(X,α)

andK1(C∗(X × T, α × ϕ)) ∼= Z ⊕ K0(X × Z2, α × o(ϕ))/K0(X, α),

where the Z-summand of the K1-group is generated by the implementing unitary.

By the same reason as orientation preserving cases, one sees that K0(C∗(X × T, α × ϕ)) isunital order isomorphic to K0(X, α), when α × ϕ is minimal.

We remark that the torsion subgroup of

K0(X × Z2, α × o(ϕ))/K0(X,α)

is isomorphic to Z2 by [M3, Lemma 4.5]. The torsion element is given by

f0(x, k) =

{1 o(ϕ)(α−1(x)) = 1 and k = 00 otherwise.

3 Real rank and stable rank

Let (X,α) be a Cantor minimal system and let X ∋ x 7→ ϕx ∈ Homeo(T) be a continuous map.In this section, we would like to compute the real rank and the stable rank of C∗(X ×T, α×ϕ).Let us denote the crossed product C∗-algebra by A and the implementing unitary by u. Ourcrucial tool is a “large” subalgebra of A, which is described below. The proof will be done byapplying the argument of [Ph2] to that subalgebra.

We would like to begin with the definition of the rigidity.

Definition 3.1. Let α × ϕ be a homeomorphism on the product of the Cantor set X and thecircle T. We say that α × ϕ is rigid, if the canonical factor map from (X × T, α × ϕ) to (X, α)induces an isomorphism between the sets of invariant probability measures.

Remark 3.2. In the definition above, even if α is minimal and α×ϕ is rigid, α×ϕ may not beminimal. For example, let (X, α) be an odometer system and let ϕ ∈ Homeo+(T) be a Denjoyhomeomorphism, that is, the rotation number of ϕ is irrational but ϕ is not conjugate to arotation. It is well-known that ϕ has a unique invariant nontrivial closed subset Y , and so α×ϕhas a nontrivial closed invariant subset X ×Y . Thus α×ϕ is not minimal. It is also known thatY is the Cantor set and ϕ|Y is minimal. This Cantor minimal system is called a Denjoy system(see [PSS] for details). The complement of Y consists of countable disjoint open intervals andeach interval is a wandering set. Hence, for every α × ϕ-invariant probability measure µ, wehave µ(X ×Y c) = 0. Moreover, as pointed out in [M1, Section 7 (2)], the product of (X,α) and(Y, ϕ|Y ) is uniquely ergodic. It follows that α × ϕ is uniquely ergodic. In particular, it is rigid.

For x ∈ X, let Ax be the C∗-subalgebra generated by C(X ×T) and uC0((X \ {x})×T). In[Pu1, Theorem 3.3], it was proved that Ax ∩ C∗(X,α) is a unital AF algebra, where we regardC∗(X,α) as a C∗-subalgebra of A. This AF subalgebra played a crucial role in the subsequentpapers [HPS] and [GPS]. In our situation, the C∗-subalgebra Ax is not an AF algebra but anAT algebra, and it helps us to show real rank zero.

Proposition 3.3. In the setting above, we have the following.

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(1) Ax is a unital AT algebra.

(2) When α×ϕ is orientation preserving, K0(Ax) is unital order isomorphic to K0(X,α) andK1(Ax) is isomorphic to K0(X,α).

(3) When α×ϕ is not orientation preserving, K0(Ax) is unital order isomorphic to K0(X, α)and K1(Ax) is isomorphic to an extension of Coker(id−α∗

ϕ) by Z.

(4) There exists a canonical bijection between the tracial state space T (Ax) and the set ofα × ϕ-invariant probability measures.

(5) Ax is simple if and only if α × ϕ is minimal.

(6) If α × ϕ is minimal, then Ax is real rank zero if and only if α × ϕ is rigid.

Proof. (1) LetPn = {X(n, v, k) : v ∈ Vn, k = 1, 2, . . . , h(v)}

be a sequence of Kakutani-Rohlin partitions which gives a Bratteli-Vershik model for (X,α) (see[HPS, Theorem 4.2] or [M3, Section 2] for Kakutani-Rohlin partitions). We assume that thesequence of the roof sets

R(Pn) =⋃

v∈V

X(n, v, h(v))

shrinks to a singleton {x}. Let An be the C∗-subalgebra generated by C(X×T) and uC(R(Pn)c×T). It is easy to see that Ax is the norm closure of the union of all An’s. By using a similarargument to [Pu1, Lemma 3.1], it can be shown that An is isomorphic to

⊕

v∈Vn

Mh(v) ⊗ C(X(n, v, h(v))) ⊗ C(T),

which is an AT algebra. To verify this, define a projection pv by

pv =h(v)∑

k=1

1X(n,v,k)

for each v ∈ V . Since pvu(1 − 1R(Pn)) = u(1 − 1R(Pn))pv, the projection pv is central in An.Clearly ui−j1X(n,v,j) for i, j = 1, 2, . . . , h(v) form matrix units of pvAnpv. By

1X(n,v,h(v))An1X(n,v,h(v)) = C(X(n, v, h(v)) × T),

we obtain the description above. Hence Ax is also an AT algebra.(2) There is a natural homomorphism from Ki(C(X ×T)) ∼= C(X, Z) to Ki(An) for i = 1, 2.

By (1), the kernel of this map is

{f − f ◦ α−1 : f ∈ C(X, Z), f(y) = 0 for all y ∈ R(Pn)}.

Therefore Ki(Ax) is isomorphic to

C(X, Z)/{f − f ◦ α−1 : f ∈ C(X, Z), f(x) = 0}.

It follows from 1X ◦ α−1 = 1X that

{f − f ◦ α−1 : f ∈ C(X, Z), f(x) = 0} = {f − f ◦ α−1 : f ∈ C(X, Z)},

which implies Ki(Ax) ∼= K0(X, α).

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(3) The computation of K0-group is the same as the orientation preserving case. Let usconsider K1(Ax). It is not hard to see that K1(Ax) is isomorphic to

C(X, Z)/{f − α∗ϕ(f) : f ∈ C(X, Z) and f(x) = 0}.

We follow the notation used in the discussion before Lemma 2.5. The image of

{f − α∗ϕ(f) : f ∈ C(X, Z) and f(x) = 0}

by δ is equal to the image of

{f − f ◦ (α × o(ϕ))−1 : f ∈ C(X × Z2, Z), f(x, 0) = (x, 1) = 0}

by id−γ∗. Hence, in the same way as Lemma 2.5, we have

K1(Ax) ∼= K0(X × Z2, α × o(ϕ); Z2)/K0(X,α).

See [M2] for the definition of K0(X × Z2, α × o(ϕ); Z2). By [Pu1, Theorem 4.1],

0 → Z → K0(X × Z2, α × o(ϕ); Z2) → K0(X × Z2, α × o(ϕ)) → 0

is exact, which implies that K1(Ax) is an extension of

Coker(id−α∗ϕ) ∼= K0(X × Z2, α × o(ϕ))/K0(X, α)

by the integers Z.(4) Since there exists a canonical bijection between T (A) and the set of α × ϕ-invariant

probability measures, it suffices to check that every τ ∈ T (Ax) extends to a tracial state on A.Let f ∈ C(X ×T) be a function satisfying 0 ≤ f ≤ 1. Take a natural number n ∈ N arbitrarily.We can find a clopen neighborhood U of x such that U,α−1(U), . . . , α−n(U) are mutually disjoint.Let p = 1U×T ∈ C(X ×T). Then we have τ(p) < n−1, because p, u∗pu, . . . , un∗pun are mutuallyequivalent in Ax and mutually disjoint. We also notice that u(1− p)f belongs to Ax. It followsthat

τ(f) ≤ τ(p) + τ((1 − p)f) = τ(p) + τ(u(1 − p)fu∗) <1n

+ τ(ufu∗).

Similarly we can see τ(ufu∗) < n−1 + τ(f). Since n is arbitrary, τ(f) equals τ(ufu∗), whichmeans that τ extends to a trace on A.

(5) Note that the C∗-algebra A can be regarded as a groupoid C∗-algebra associated withthe equivalence relation

R = {(z, (α × ϕ)k(z)) : z ∈ X × T, k ∈ Z}.

Then the C∗-subalgebra Ax corresponds to the subequivalence relation

Rx = R \ {((α × ϕ)k(x, t), (α × ϕ)l(x, t)) : t ∈ T, (1 − k, l) ∈ N2 or (k, 1 − l) ∈ N2}.

It is well-known that a homeomorphism on a compact space is minimal if and only if everypositive orbit is dense. Therefore α×ϕ is minimal if and only if each equivalence class of Rx isdense in X × T. It follows from [R, Proposition 4.6] that this is equivalent to Ax being simple.

(6) By (1) and (5), Ax is a unital simple AT algebra. By (2) and (3), in the K0-group, everyprojection of Ax is equivalent to some [f ] ∈ K0(X, α). Hence, projections in Ax separate traceson Ax if and only if α×ϕ is rigid. Then the conclusion follows from [BBEK, Theorem 1.3].

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Remark 3.4. The six-term exact sequence of [Pu2, Theorem 2.4] applies to this situation. See[Pu2, Example 2.6]. The reduced groupoid C∗-algebra C∗

r (H) appearing there is isomorphic toC(T) ⊗K.

We would like to consider the real rank of A. In [Ph2], it was shown that if G is an almostAF Cantor groupoid and C∗

r (G) is simple, then C∗r (G) has real rank zero. The key of its proof

was the presence of a “large” AF subalgebra. We will show that a similar argument is possiblewhen the AF subalgebra is replaced by a subalgebra with tracial rank zero.

Definition 3.5 ([L5, Theorem 6.13]). We say that a unital simple C∗-algebra A has tracial(topological) rank zero, if for any finite subset F ⊂ A, any ε > 0 and any nonzero positiveelement c ∈ A, there exists a projection e ∈ A and a finite dimensional unital subalgebraE ⊂ eAe (that is, e is the identity of E) such that:

(1) ∥ae − ea∥ < ε for all a ∈ F .

(2) For every a ∈ F , there is b ∈ E such that ∥pap − b∥ < ε.

(3) 1 − e is Murray-von Neumann equivalent to a projection in cAc.

Definition 3.6. A unital simple C∗-algebra A is called an almost tracially AF algebra, if thereexists a unital simple subalgebra B ⊂ A with tracial rank zero such that the following holds:for any finite subset F ⊂ A and any ε > 0, there exists a projection p ∈ B such that:

(1) For every a ∈ F , there is b ∈ B such that ∥ap − b∥ < ε.

(2) τ(1 − p) < ε for every tracial state τ ∈ T (B).

Lemma 3.7. Suppose that α × ϕ is a minimal homeomorphism on X × T. If α × ϕ is rigid,then the crossed product C∗-algebra A = C∗(X × T, α × ϕ) is an almost tracially AF algebra.

Proof. Take x ∈ X. By Proposition 3.3, Ax is a unital simple AT algebra of real rank zero.It follows from [L1, Proposition 2.6] that Ax is a unital simple algebra with tracial rank zero.Suppose that a finite subset F ⊂ A and ε > 0 are given. Since

{N∑

k=−N

fkuk : N ∈ N, fk ∈ C(X × T)

}

is a dense subalgebra of A, we may assume that there is N ∈ N such that F is contained in

EN =

{N∑

k=−N

fkuk : fk ∈ C(X × T)

}.

We can find a clopen neighborhood U of x so that α1−N (U), α2−N (U), . . . , U, α(U), . . . , αN (U)are mutually disjoint and µ(U) < ε/2N for all µ ∈ Mα. Put V =

⋃Nk=1−N αk(U) and p =

1V c×T ∈ C(X ×T). It is easy to see that ukp and u1−kp belong to Ax for k = 0, 1, . . . , N , and soap ∈ Ax for every a ∈ EN . Moreover, τ(1 − p) is less than ε for every τ ∈ T (Ax). This finishesthe proof of the lemma.

The following lemma is a generalization of [Ph2, Lemma 4.3].

9

Lemma 3.8. Let A be a unital simple algebra with tracial rank zero. Let p ∈ A be a projectionand let a ∈ A be a nonzero self-adjoint element. For any ε > 0 and n ∈ N satisfying nε > 1,there exists a projection q ∈ A such that ∥qa − aq∥ < ε∥a∥, p ≤ q and

τ(q) < (2n + 1)τ(p)

for all τ ∈ T (A).

Proof. Without loss of generality, we may assume ∥a∥ = 1. Put ε0 = 8−1(ε − n−1). Chooseδ0 > 0 so that whenever p, q ∈ A are projections satisfying ∥pq − p∥ < δ0, then there exists aunitary u ∈ A such that

∥u − 1∥ < ε0 and p ≤ uqu∗.

Letδ = min{ε0, 4−1δ0, (2n + 1)−1τ(p) : τ ∈ T (A)}.

By [L1, Proposition 2.4] together with results of [L1, Section 3], there exist a projection e ∈ Aand a finite dimensional unital subalgebra E ⊂ eAe such that:

• ∥ae − ea∥ < δ and ∥pe − ep∥ < δ.

• There exist b, c ∈ E such that ∥eae − b∥ < δ and ∥epe − c∥ < δ.

• τ(1 − e) < δ for every tracial state τ ∈ T (A).

We may assume that b = b∗, ∥b∥ = 1 and c is a projection. Thanks to [Ph2, Lemma 4.2], thereexists a projection q0 ∈ E such that

c ≤ q0, [q0] ≤ 2n[c] ∈ K0(E), and ∥q0b − bq0∥ <1n

.

Put q = 1 − e + q0. Since

∥pq − p∥≤ ∥(p(1 − e) + peq0) − (p(1 − e) + pcq0)∥ + ∥(p(1 − e) + pcq0) − (p(1 − e) + pe)∥< 2δ + 2δ ≤ δ0,

there is a unitary u ∈ A such that ∥u − 1∥ < ε0 and p ≤ uqu∗. It is not hard to see that

τ(uqu∗) = τ(q) = τ(1 − e) + τ(q0)< δ + 2nτ(c) < (2n + 1)δ + 2nτ(pe) < (2n + 1)τ(p)

for all τ ∈ T (A) and that

∥[uqu∗, a]∥ < ∥[q, a]∥ + 4ε0

< ∥[q, b]∥ + 4δ + 4ε0

<1n

+ 8ε0 = ε,

thereby completing the proof.

The following is a well-known matrix trick. We omit the proof.

Lemma 3.9. When a is a self-adjoint element of a unital C∗-algebra A,[a 00 0

]

is approximated by an invertible self-adjoint element in A ⊗ M2.

10

Although the proof of the following theorem is almost the same as that of [Ph2, Theorem4.7], we would like to state it for the reader’s convenience.

Theorem 3.10. If a unital simple C∗-algebra A is an almost tracially AF algebra, then A hasreal rank zero.

Proof. Let B ⊂ A be a unital simple subalgebra with tracial rank zero as in Definition 3.6.Let a ∈ A be self-adjoint and non-invertible. It suffices to show that a is approximated by a

self-adjoint invertible element of A. Without loss of generality, we may assume ∥a∥ ≤ 1. Takeε > 0 arbitrarily. Define a continuous function g on [−1, 1] by

g(t) =

{1 − ε−1|t| |t| ≤ ε

0 otherwise.

Putε0 = min{τ(g(a)) : τ ∈ T (A)}.

Since A is simple, ε0 is positive. Applying [Ph2, Lemma 4.4] to g : [−1, 1] → [0, 1] and 4−1ε0 > 0,we obtain δ > 0. We may assume that δ is less than ε. Choose a natural number n ∈ N so that

1n

< min{

ε0

12,δ

2

}.

By definition, there is a projection p ∈ B such that a(1 − p) is close to B and τ(p) is less thann−2 for all τ ∈ T (B). By perturbing a, we may assume that a(1 − p) belongs to B. Then, wecan apply Lemma 3.8 to a − pap ∈ B and p ∈ B, and get a projection q ∈ B such that p ≤ q,

∥[q, a − pap]∥ <δ

2∥a − pap∥ ≤ δ

andτ(q) ≤ (2n + 1)τ(p) <

3n

<ε0

4for every tracial state τ ∈ T (B). It follows that

τ(g(a)) − τ(q) > ε0 −ε0

4>

ε0

4.

We also notice that ∥[q, a]∥ < δ. Put a0 = (1 − q)a(1 − q) ∈ B. By the choice of δ, we have

τ(g(a0)) > τ(g(a)) − τ(q) − ε0

4>

ε0

2.

A unital simple algebra with tracial rank zero has real rank zero by [L1, Theorem 3.4], and sothere exists a projection r in the hereditary subalgebra of B generated by g(a0) such that

∥rg(a0) − g(a0)∥ <ε0

4.

Thenτ(r) ≥ τ(rg(a0)r) > τ(g(a0)) −

ε0

4>

ε0

4for all τ ∈ T (B). Since the order on projections of B is determined by traces (see [L2, Theorem6.8, 6.13]), there is a projection r0 ∈ B such that r0 ≤ r and r0 ∼ q. Moreover, by means of[Ph2, Lemma 4.6], we have

∥r0a0 − a0r0∥ < 2ε and ∥r0a0r0∥ < ε.

11

As a result,

a2ε≈ qaq + a0

4ε≈ qaq + r0a0r0 + (1 − q − r0)a0(1 − q − r0)ε≈ qaq + (1 − q − r0)a0(1 − q − r0)

is obtained. The element (1 − q − r0)a0(1 − q − r0) belongs to B and B has real rank zero. Byapplying Lemma 3.9 to qaq and r0 ∼ q, we can get the conclusion.

Corollary 3.11. For a minimal homeomorphism α×ϕ on X ×T, the following are equivalent.

(1) α × ϕ is rigid.

(2) The crossed product C∗-algebra A = C∗(X × T, α × ϕ) has real rank zero.

(3) D(K0(A)) is uniformly dense in Aff(T (A)).

Proof. (1)⇒(2). This is immediate from Lemma 3.7 and the theorem above.(2)⇒(3). Since α × ϕ is minimal, A is simple. By Theorem 3.12, A is stably finite and the

projections in A ⊗K satisfy cancellation. Furthermore, K0(A) is weakly unperforated by [Ph1,Theorem 4.5]. It follows from [B, Theorem 6.9.3] that the image of K0(A) is uniformly densein real valued affine continuous functions on QT (A), the set of quasitraces on A. Because everyelement of Aff(T (A)) comes from a self-adjoint element of A (see [BKR, Proposition 3.12] forinstance), it extends to a real valued affine continuous function on QT (A). Hence D(K0(A)) isuniformly dense in Aff(T (A)). Note that if one uses the deep result obtained by Haagerup in[H], the latter half of the proof is superfluous.

(3)⇒(1). Suppose that α×ϕ is not rigid. Thus, there are ν1, ν2 ∈ Mα×ϕ such that F∗(ν1) =F∗(ν2) = ν ∈ Mα, where F is the canonical factor map onto (X, α). Let τ1 and τ2 be the tracialstates on A arising from ν1 and ν2. Since

τ1∗([f ]) = ν1(f ◦ F ) = ν(f) = ν2(f ◦ F ) = τ2∗([f ])

for all [f ] ∈ K0(X,α), projections in C∗(X,α) cannot separate traces on A. Therefore, we canfinish the proof here when α × ϕ is not orientation preserving.

Assume that α × ϕ is orientation preserving. By Lemma 2.4, K0(A) is isomorphic to Z ⊕K0(X, α). Suppose that there are projections e1, e2 ∈ Mk(A) for some integer k such that[e1]− [e2] = (1, 0) ∈ Z⊕K0(X, α). Then, for any x = (n, [f ]) ∈ K0(A) ∼= Z⊕K0(X, α), we have

τ1∗(x) − τ2∗(x) = n(τ1(e1 − e2) − τ2(e1 − e2)).

Hence{τ1∗(x) − τ2∗(x) : x ∈ K0(A)}

is discrete in R, which is a contradiction.

We now turn to a consideration of stable rank of A. Suppose that α×ϕ is minimal but maynot be rigid. The C∗-subalgebra Ax may not have real rank zero. But, it is still a unital simpleAT algebra by Proposition 3.3. Moreover, Lemma 3.7 is also valid when one replaces ‘almosttracially AF’ by ‘almost AT’. A unital simple AT algebra is known to have property (SP), thatis, every nonzero hereditary subalgebra contains a nonzero projection. Hence we see that A alsohas property (SP) by virtue of Lemma 3.7. We also remark that Ax has stable rank one and theorder on projections of Ax is determined by traces, because Ax is a unital simple AT algebra.

12

Then, by reading the proof of [Ph2, Theorem 5.2] carefully, it turns out that A and the “large”subalgebra Ax do not need to have real rank zero and that they only need to have property (SP)so that the proof works. As a consequence, we have the following.

Theorem 3.12. When α × ϕ is a minimal homeomorphism on X × T, the crossed productC∗-algebra A = C∗(X × T, α × ϕ) has stable rank one. In particular, the projections in A ⊗ Ksatisfy cancellation.

In [Ph1, Theorem 4.5], it was proved that A satisfies the K-theoretic version of Blackadar’ssecond fundamental comparability question, that is, if x ∈ K0(A) satisfies τ∗(x) > 0 for allτ ∈ T (A), then x ∈ K0(A)+. In particular, K0(A) is weakly unperforated. Combining this withthe theorem above, we can deduce the following.

Theorem 3.13. When α×ϕ is a minimal homeomorphism on X ×T, the order on projectionsof M∞(C∗(X × T, α × ϕ)) is determined by traces. In other words, C∗(X × T, α × ϕ) satisfiesBlackadar’s second fundamental comparability question.

4 Cocycles with values in the rotation group

Let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map. In this section,we would like to investigate a homeomorphism α×Rξ on X ×T and its related crossed productC∗-algebra A = C∗(X × T, α × Rξ). Of course, α × Rξ is orientation preserving.

Definition 4.1. Let α be a minimal homeomorphism on X. Define

K0T(X, α) = C(X, T)/{η − ηα−1 : η ∈ C(X, T)}.

The equivalence class of ξ ∈ C(X, T) in K0T(X,α) is denoted by [ξ]α or [ξ].

Let θ ∈ T and put ξ(x) = θ for all x ∈ X. Thus, ξ is a constant function. It is easy to seethat [ξ] is zero in K0

T(X,α) if and only if θ is a topological eigenvalue of (X, α). The reader mayrefer to [W, Theorem 5.17] for topological eigenvalues. Since X is compact, the set of topologicaleigenvalues is at most countable. It follows that K0

T(X,α) is uncountable.At first, we describe when α × Rξ is minimal in terms of K0

T(X, α). Note that more generalresults were obtained in [Pa].

Lemma 4.2. Let (X, α) be a Cantor minimal system and ξ : X → T be a continuous map.Then, α × Rξ is minimal if and only if n[ξ] = 0 in K0

T(X,α) for all n ∈ N.

Proof. Suppose that there exist n ∈ N and η such that nξ = η − ηα−1. Then

{(x, t) ∈ X × T : nt = η(α−1(x))}

is closed and invariant under α × Rξ, and so α × Rξ is not minimal.Let us prove the other implication. Assume that α×Rξ is not minimal. Let E be a minimal

subset of α ×Rξ. Note that id×Rt commutes with α ×Rξ. Since E is not the whole of X × T,

G = {t ∈ T : (id×Rt)(E) = E}

is a closed proper subgroup of T. It follows that there exists n ∈ N such that G = {t ∈ T : nt =0}. Moreover, on account of the minimality of E, we can deduce that there exists η : X → Tsuch that

E = {(x, t) ∈ X × T : nt = η(x)}.

The map η is continuous, because E is closed. Hence we have nξ = ηα − η.

13

If α × Rξ is not minimal, then there exist uncountably many minimal closed subsets. Inparticular, it is not rigid. Compare this with Remark 3.2.

Lemma 4.3. Let (X, α) and (Y, β) be Cantor minimal systems. Let ξ and ζ be continuousmaps from X to T. Suppose that α × Rξ is minimal. Then, α × ξ and β × ζ is conjugate ifand only if there exists a homeomorphism F : X → Y such that Fα = βF , and [ξ]α = [ζF ]α or[ξ]α = −[ζF ]α in K0

T(X,α).

Proof. The ‘if’ part is clear. We consider the ‘only if’ part. Let F × ϕ : X × T → Y × T be aconjugating map, that is,

Fα = βF and ϕα(x)(s + ξ(x)) = ϕx(s) + ζ(F (x))

for all (x, s) ∈ X×T. For every t ∈ T, id×Rt commutes with β×Rζ , and so (F×ϕ)−1(id×Rt)(F×ϕ) commutes with α × Rξ. Let x ∈ X and put s = ϕ−1

x (ϕx(0) + t). Then we have

(F × ϕ)−1(id×Rt)(F × ϕ)(x, 0) = (x, s) = (id×Rs)(x, 0).

By the minimality of α × ξ, we can conclude that

(F × ϕ)−1(id×Rt)(F × ϕ) = id×Rs.

It follows that the mapping t 7→ s is a continuous injective homomorphism from T to T. Thus,

(F × ϕ)−1(id×Rt)(F × ϕ) = id×Rt

for all t ∈ T, or(F × ϕ)−1(id×Rt)(F × ϕ) = id×R−t

for all t ∈ T. Without loss of generality we may assume the first, which yields

ϕx(s + t) = ϕx(s) + t

for all (x, s) ∈ X × T and t ∈ T. It follows that ϕx equals Rϕx(0) and

ξ(x) + ϕα(x)(0) = ϕx(0) + ζ(F (x)).

Thereby the assertion follows.

Next, we would like to consider when α × Rξ is rigid. Although the following is a specialcase of [Pa, Theorem 3] or [Z, Theorem 3.5], we include the proof for the reader’s convenience.

Lemma 4.4. Let (X, α) be a Cantor minimal system. For a continuous function ξ : X → T,the following are equivalent.

(1) α × Rξ is rigid.

(2) Every α × Rξ-invariant measure ν is also id×Rt-invariant for all t ∈ T, that is, ν is aproduct measure of the Haar measure on T and an invariant measure for (X, α).

(3) For every α-invariant measure µ on X and n ∈ N, there does not exist a Borel functionη : X → T such that nξ(x) = η(x) − ηα−1(x) for µ-almost every x ∈ X.

14

Proof. We denote the canonical factor map from (X × T, α × Rξ) to (X,α) by π.(1)⇒(2). This is immediate from π∗ ◦ (id×Rt)∗ = (π ◦ (id×Rt))∗ = π∗.(2)⇒(1). Define a continuous map Φ : C(X × T) → C(X) by

Φ(f)(x) =∫

Tf(x, t) dt.

If ν ∈ Mα×Rξis id×Rt-invariant, then ν(f) = π∗(ν)(Φ(f)) for all f ∈ C(X × T). Hence

π−1∗ (π∗(ν)) = {ν}.

(2)⇒(3). Suppose that there exist an α-invariant measure µ ∈ Mα, n ∈ N and a Borelfunction η : X → T such that

nξ(x) = η(x) − ηα−1(x)

for µ-almost every x ∈ X. Then

C(X × T) ∋ f 7→ 1n

∫

X

∑

nt=ηα−1(x)

f(x, t) dµ(x) ∈ C

yields a probability measure on X ×T. Note that the summation runs over n distinct t’s whichsatisfy nt = ηα−1(x). This measure is α × Rξ-invariant, because we have

∑

nt=ηα−1(x)

f(α(x), t + ξ(x)) =∑

nt=ηα−1(α(x))

f(α(x), t)

for µ-almost every x ∈ X. But it is not the product of the Haar measure and µ.(3)⇒(2). Suppose that ν ∈ Mα×Rξ

is not invariant under the rotation id×Rt. We mayassume that ν is an ergodic measure. It follows from [KH, Corollary 4.1.9] or [W, Lemma 6.13]that there exists an Fσ subset E ⊂ X × T such that ν(E) = 1 and

limn→∞

1n

n−1∑

k=0

f((α × Rξ)k(x, s)) = ν(f)

for all f ∈ C(X × T) and (x, s) ∈ E. We may assume that E is α × Rξ-invariant. Put

G = {t ∈ T : (id×Rt)∗(ν) = ν}.

By assumption, G is a closed proper subgroup of T. Thus, there is n ∈ N such that G = {t ∈T : nt = 0}. If (x, s) belongs to E, then we have

limn→∞

1n

n−1∑

k=0

f((α × Rξ)k(x, s + t)) = ν(f ◦ (id×Rt)) = (id×Rt)∗(ν)(f)

for all f ∈ C(X × T). Therefore (id×Rt)(E) ∩ E is empty if t /∈ G. Furthermore, by replacingE by ⋃

t∈G

(id×Rt)(E),

we may assume that (id×Rt)(E) = E for all t ∈ G. On the Fσ subset π(E) ⊂ X, we definea T-valued function η by η(x) = nt for (x, t) ∈ E. If E0 ⊂ E is closed, then η is evidentlycontinuous on π(E0). It follows that η is a well-defined Borel function. For (x, s) ∈ E,

(α × Rξ)(x, s) = (α(x), s + ξ(x))

belongs to E, and soη(α(x)) = ns + nξ(x) = α(x) + nξ(x)

is obtained. Since this equation holds for all x ∈ π(E) and π∗(ν)(π(E)) ≥ ν(E) = 1, the proofis completed.

15

Let µ ∈ Mα. As in the discussion following Definition 4.1, let ξ(x) = θ be a constant function.Then, there exists a Borel function η ∈ C(X, T) such that

ξ(x) = η(x) − ηα−1(x)

for µ-almost every x ∈ X if and only if e2π√−1θ is an eigenvalue of the unitary operator πµ(uα) ∈

L2(X, µ), where πµ is a representation of C∗(X, α) corresponding to the invariant measure µ.Since L2(X, µ) is separable, eigenvalues of a unitary operator are at most countable. Hence,by the lemma above, we can obtain a lot of rigid homeomorphisms. Moreover, it is knownthat eigenvalues of the unitary operator πµ(uα) need not be topological eigenvalues of (X, α).Therefore we can see that there exists a minimal homeomorphism α×Rξ which is not rigid. Wewill look at a concrete example in Example 9.1.

There is another way to find a rigid homeomorphism. Let ξ ∈ C(X, T) and let ξ ∈ C(X, R)be its lift (this is always possible because X is the Cantor set). Then µ 7→ µ(ξ) gives anaffine function from the set of α-invariant measures Mα to R. By the lemma above, if nµ(ξ) /∈µ(C(X, Z)) for each ergodic α-invariant measure µ and n ∈ N, then α × Rξ is rigid.

Next, we will show that A = C∗(X × T, α × Rξ) can be written as a crossed product ofC∗(X,α) by a certain action. Define an automorphism ι(ξ) on C∗(X,α) by ι(ξ)(f) = f for allf ∈ C(X) and ι(ξ)(uα) = uαe2π

√−1ξ(x), where uα denotes the implementing unitary of C∗(X, α).

This kind of automorphism was considered in [M1]. We remark that ι(ξ) is approximately inner,because ι(ξ)∗ is the identity on the K-groups (or one can deduce it from Lemma 6.1 or [M1,Lemma 5.1]). Let ι(ξ) denote the dual action on C∗(X, α) oι(ξ) Z.

Proposition 4.5. There is an isomorphism π from the crossed product C∗-algebra A = C∗(X×T, α × Rξ) to C∗(X,α) oι(ξ) Z such that π(f) = f for all f ∈ C(X) and π(g ◦ (id×Rt)) =ι(ξ)t(π(g)) for all g ∈ C(X × T) and t ∈ T.

Proof. In order to avoid confusion, we have to use different symbols for three implementingunitaries: we denote the implementing unitary in C∗(X,α) by uα and denote the unitary im-plementing ι(ξ) by v, while u ∈ A denotes the unitary implementing α × Rξ.

Let z ∈ C(X × T) be a unitary defined by z(x, t) = e2π√−1t. Define π(z) = v and π(f) = f

for all f ∈ C(X) ⊂ C(X × T). This is well-defined because v and f commute in C∗(X, α) o Z.Moreover, π is an isomorphism from C(X × T) onto its image. We define π(u) = uα. It is easyto check that

π(u)π(f)π(u∗) = fα−1 = π(fα−1)

for all f ∈ C(X) and that

π(u)π(z)π(u∗) = e−2π√−1ξ(α−1(x))v = π(e−2π

√−1ξ(α−1(x))z) = π(z ◦ α−1).

Therefore π is a homomorphism from A to C∗(X,α) o Z. Clearly π is surjective. It is alsostraightforward to see π(g ◦ (id×Rt)−1) = ι(ξ)t(π(g)) for all g ∈ C(X × T) and t ∈ T.

It remains to show that π is an isomorphism. Let E be the conditional expectation from Ato C(X ×T). It is well-known that E is faithful. We can define an action of T on C∗(X, α)×Zby

γt(f) = f, γt(uα) = e2π√−1tuα, and γt(v) = v

for t ∈ T. LetE0(a) =

∫

Tγt(a) dt

for a ∈ C∗(X,α) o Z. Then we have π ◦ E = E0 ◦ π. The faithfulness of E leads us to theconclusion.

16

Remark 4.6. By [Pu1, Corollary 5.7] or [HPS, Theorem 5.5] there exist bijective correspon-dences between the following spaces.

(1) The state space S(K0(X, α)) of K0(X, α).

(2) The tracial state space T (C∗(X, α)) of C∗(X,α).

(3) The set Mα of all α-invariant probability measures.

By Proposition 4.5 and Lemma 4.4, (2) and (3) of the above are also identified with the following.

(4) The set of ι(ξ)-invariant traces on A = C∗(X × T, α × Rξ).

(5) The set of probability measures on X×T which are invariant under α×Rξ and the rotationid×Rt.

Suppose that s ∈ S(K0(A)) is a state on the ordered group K0(A). Then there is ν ∈ Mα

such that s([f ]) = Sν([f ]) for all [f ] ∈ K0(X, α), where Sν is a state on K0(X,α) coming fromν and K0(X, α) is viewed as a subgroup of K0(A). The α-invariant measure ν extends to anα × Rξ-invariant measure on X × T, and so we can extend Sν on K0(A) (different choices ofα × Rξ-invariant measures do not concern the extension of Sν). For x ∈ K0(A),

S(K0(X,α)) ∋ Sµ 7→ Sµ(x) ∈ R

is an affine function on the state space S(K0(X,α)). Since the image of K0(X, α) is dense inAff(S(K0(X, α))), for any ε > 0, there exists f1, f2 ∈ C(X, Z) such that

Sµ(x) − ε < Sµ([f1]) < Sµ(x) < Sµ([f2]) < Sµ(x) + ε

for all µ ∈ Mα. If A is simple, then it follows from Theorem 3.13 that [f1] < x < [f2] in K0(A).In particular, we have

Sν(x) − ε < Sν([f1]) = s([f1]) < s(x) < s([f2]) = Sν([f2]) < Sν(x) + ε.

Hence s is equal to Sν as a state on K0(A). Consequently, the state space S(K0(X,α)) can beidentified with

(6) The state space S(K0(A)) of A

when A is simple.

Theorem 4.7. Let (X, α) be a Cantor minimal system and let ξ ∈ C(X, T). Suppose that α×Rξ

is a minimal homeomorphism on X×T. For the unital simple C∗-algebra A = C∗(X×T, α×Rξ),the following conditions are equivalent.

(1) α × Rξ is rigid.

(2) A has real rank zero.

(3) For every extremal tracial state τ on C∗(X,α) and every n ∈ N, ι(ξ)n is not weakly innerin the GNS representation πτ .

If C∗(X, α) has finitely many extremal traces, then the conditions above are also equivalent to

(4) ι(ξ) has the tracial cyclic Rohlin property.

17

Proof. (1)⇔(2) was shown in Corollary 3.11.(1)⇒(3) follows [K, Proposition 2.3].(3)⇒(1). Suppose that α × Rξ is not rigid. By Lemma 4.4, there exist an ergodic measure

µ ∈ Mα, n ∈ N and a Borel function η : X → T such that

nξ(x) = η(x) − ηα(x)

for µ-almost every x ∈ X. Define h ∈ L∞(X, µ) by h(x) = e2π√−1η(x) and let V be the

multiplication operator by h on L2(X,µ). Let τ be the extremal trace on C∗(X,α) correspondingto µ. We can regard πτ (C∗(X, α)) as a C∗-subalgebra of B(L2(X, µ)). Then, it is not hard tosee that V commutes with πτ (f) for all f ∈ C(X) and that

V ∗πτ (uα)V = πτ (ι(ξ)n(uα)).

Namely ι(ξ)n is weakly inner in the GNS representation πτ .(3)⇒(4). From [OP] we can see that ι(ξ) has the tracial Rohlin property. The conclusion

follows from [LO, Theorem 3.4].(4)⇒(2). It follows from [LO, Theorem 2.9] that A has tracial rank zero. The conclusion is

immediate from [L1, Theorem 3.4].

In the theorem above, it was shown that if α×Rξ is rigid and α has only finitely many ergodicmeasures, then A has tracial rank zero. In the next section we will prove that the hypothesis offinitely many ergodic measures is actually not necessary.

5 Tracial rank

Throughout this section, let (X,α) be a Cantor minimal system and let ξ : X → T be acontinuous map. We denote the crossed product C∗-algebra C∗(X × T, α × Rξ) by A and itsimplementing unitary by u. We would like to show that if α × Rξ is rigid then A has tracialrank zero. The proof will be done by some improvement of Lemma 3.7. Following the notationused there, we define Ax = C∗(C(X × T), uC0((X \ {x}) × T)) for x ∈ X.

Define z ∈ C(X × T) by z(x, t) = e2π√−1t. The key step of the proof is approximately

unitary equivalence of z1U and z1V in Ax, where U and V are suitable clopen subsets of Xsatisfying [1U ] = [1V ]. When one uses the fact that Ax has tracial rank zero, the proof is justan application of [L6, Theorem 3.4], in which actually more general result has been obtained.But we would like to include an elementary proof which does not use tracial rank zero for thereader’s convenience.

The following lemma says that rigidity implies that the values of a cocycle are uniformlydistributed in T.

Lemma 5.1. Suppose that α×Rξ is rigid. For any irrational s ∈ T and any ε > 0, there existsN ∈ N such that the following is satisfied: for any n ≥ N and y ∈ X there is a permutation σon {1, 2, . . . , n} such that

|ks −σ(k)−1∑

i=0

ξ(αi(y))| < ε

holds for all k ∈ {1, 2, . . . , n}.

Proof. For n ∈ N, put ξn(x) =∑n−1

i=0 ξ(αi(x)). By Lemma 4.4, we have∫

X×Tf(t) dν =

∫

Tf(t) dt

18

for every invariant measure ν of α×Rξ and every f ∈ C(T). Hence for any f ∈ C(T) and ε > 0there exists N ∈ N such that

∣∣∣∣∣1n

n−1∑

i=0

f(ξn(x)) −∫

Tf(t) dt

∣∣∣∣∣ < ε

for all n ≥ N and x ∈ X. By a slight modification of [KK, Lemma 2], we can get the conclusion.We leave the details to the reader.

In the following lemmas we need the idea of induced transformations. Let U be a clopensubset of X. Define r : U → N by

r(x) = min{n ∈ N : αn(x) ∈ U}.

Since α is minimal, r is well-defined and continuous. Put α(y) = αr(y)(y) for every y ∈ U . Thusα is the first return map on U . It is well-known that (U, α) is a Cantor minimal system andthe associated crossed product C∗(U, α) is canonically identified with 1UC∗(X,α)1U . Defineξ : U → T by

ξ(y) =r(y)−1∑

i=0

ξ(αi(y))

for all y ∈ U . Then α×Rξ is the first return map of α×Rξ on U ×T and the associated crossedproduct C∗(U × T, α × Rξ) is identified with 1U×TA1U×T. Note that the unitary implementingα × Rξ is given by ∑

n∈Nun1Un×T,

where Un = r−1(n) and the summation is actually finite.In general there is a bijective correspondence between invariant measures of the induced

transformation and those of the original one. It follows that α×Rξ is rigid if and only if α×Rξis rigid.

For x ∈ X, let k be the minimal natural number such that α−k(x) ∈ U and set x = α−k(x).Then it is not hard to see that

1UAx1U = C∗(C(U × T), uC0((U \ {x}) × T)).

Lemma 5.2. Let α×Rξ be a rigid homeomorphism and let x ∈ X. Suppose that U is a nonemptyclopen subset of X. For any s ∈ T and any ε > 0, there exists a unitary w ∈ 1U (Ax∩C∗(X,α))1U

such that∥wzw∗ − e2π

√−1sz1U∥ < ε.

Proof. At first we consider the case U = X. Clearly we may assume that s is irrational. Byapplying Lemma 5.1 we can find N ∈ N. Let

P = {X(v, k) : v ∈ V, k = 1, 2, . . . , h(v)}

be a Kakutani-Rohlin partition such that the roof set R(P) contains x and h(v) is greater thanN for every v ∈ V . By dividing each tower if necessary, we may assume that P is sufficientlyfiner so that whenever y1, y2 ∈ X(v, k) we have |ξ(y1) − ξ(y2)| < ε/h(v).

For every v ∈ V , choose yv ∈ X(v, 1) arbitrarily. By Lemma 5.1 there is a permutation σv

on {1, 2, . . . , h(v)} such that

|ks −σv(k)−1∑

i=0

ξ(αi(yv))| < ε

19

for all k ∈ {1, 2, . . . , h(v)}. Put

w =∑

v∈V

h(v)∑

i=1

1X(v,σv(i))uσv(i)−σv(i+1),

where u is the implementing unitary of C∗(X,α). It is easily verified that w is a unitary ofAx ∩ C∗(X, α). Moreover we get the estimate

∥wzw∗ − e2π√−1sz∥ < 4πε.

Let us consider the general case. We follow the notation used in the discussion before thelemma. Applying the first part of the proof to α × ξ and x, we obtain a unitary w in

C∗(C(U), uC0(U \ {x})) = 1U (Ax ∩ C∗(X, α))1U

which satisfies the required inequality.

Lemma 5.3. Let α × Rξ be a rigid homeomorphism and let x ∈ X. For any η ∈ C(X, T) andany ε > 0, there exists a unitary w ∈ Ax ∩ C∗(X, α) such that

∥wzw∗ − e2π√−1ηz∥ < ε,

where z ∈ C(X × T) is given by z(x, t) = e2π√−1t.

Proof. Let P be a partition of X such that whenever y1, y2 ∈ U ∈ P we have |η(y1)−η(y2)| < ε.For every U ∈ P, by Lemma 5.2, we obtain a unitary wU ∈ 1U (Ax ∩ C∗(X,α))1U satisfying

∥wUzw∗U − e2π

√−1η(x)z1U∥ < ε.

Let w be the product of all wU ’s. Then w is the desired unitary.

Lemma 5.4. Let α×Rξ be a rigid homeomorphism and let x ∈ X. Suppose that U is a clopenneighborhood of x and U,α(U), . . . , αM (U) are mutually disjoint. Put p = 1U and q = 1αM (U).Then for any ε > 0 there exists a partial isometry w ∈ Ax ∩ C∗(X,α) such that w∗w = p,ww∗ = q and

∥wzw∗ − zq∥ < ε.

Moreover we have u∗iwui ∈ Ax ∩ C∗(X, α) for all i = 0, 1, . . . , M − 1.

Proof. There exists a partial isometry v1 ∈ Ax ∩C∗(X,α) such that v∗1v1 = p and v1v∗1 = q. We

have v∗1zv1 = e2π√−1ηzp for some continuous function η defined on U . We consider the induced

transformation on U . Let α, ξ and x be as in the discussion before Lemma 5.2. Then Lemma5.3 applies to them and yields a unitary v2 ∈ p(Ax ∩ C∗(X,α))p satisfying

∥v2zpv∗2 − e2π√−1ηzp∥ < ε.

Then w = v1v2 satisfies∥wzw∗ − zq∥ < ε.

Since U,α(U), . . . , αM (U) are mutually disjoint, one can check that w belongs to Aαi(x) ∩C∗(X,α) for all i = 0, 1, . . . , M − 1. It follows that u∗iwui ∈ Ax ∩ C∗(X,α) for all i =0, 1, . . . , M − 1.

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Lemma 5.5. Suppose that α × Rξ is rigid. Let x ∈ X. For any N ∈ N, ε > 0 and a finitesubset F ⊂ C(X ×T), we can find a natural number M > N , a clopen neighborhood U of x anda partial isometry w ∈ Ax which satisfy the following.

(1) α−N+1(U), α−N+2(U), . . . , U, α(U), . . . , αM (U) are mutually disjoint, and µ(U) < ε/M forall α-invariant measure µ.

(2) w∗w = 1U and ww∗ = 1αM (U).

(3) u∗iwui ∈ Ax for all i = 0, 1, . . . , M − 1.

(4) ∥wf − fw∥ < ε for all f ∈ F .

Proof. Without loss of generality, we may assume F = {f1, f2, . . . , fk, z}, where fi belongs toC(X) ⊂ C(X × T). There exists a clopen neighborhood O of x such that

|fi(x) − fi(y)| < ε/2

for all y ∈ O and i = 1, 2, . . . , k. Since α is minimal, we can find M > N such that αM (x) ∈ O.Let U be a clopen neighborhood of x such that the condition (1) is satisfied and U∪αM (U) ⊂ O.Now Lemma 5.4 applies and yields a partial isometry w. It is clear that w is the desired one.

Theorem 5.6. Suppose that α × Rξ is minimal. Then the following are equivalent.

(1) α × Rξ is rigid.

(2) A = C(X × T, α × Rξ) has real rank zero.

(3) A = C(X × T, α × Rξ) has tracial rank zero.

(4) A = C(X × T, α × Rξ) is a unital simple AT-algebra with real rank zero.

Proof. It has been proved in Corollary 3.11 that (1) and (2) are equivalent.(3)⇔(4) follows the classification theorem of [L5], since A has torsion free K-theory (see

Lemma 2.4).(4)⇒(2) is obvious.(1)⇒(3). We will show the following: For any ε > 0, any finite subset F ⊂ C(X × T) and

any nonzero positive element c ∈ A, there exists a projection e ∈ Ax such that the followingconditions hold.

• ∥ae − ea∥ < ε for all a ∈ F ∪ {u}.

• For any a ∈ F ∪ {u}, there exists b ∈ eAxe such that ∥eae − b∥ < ε.

• 1 − e is equivalent to a projection in cAc.

It follows from Proposition 3.3 that eAxe is a unital simple AT-algebra with real rank zero.Therefore it has tracial rank zero (for example see [L3, Theorem 4.3.5]). Thus, if the above isproved, by [HLX, Theorem 4.8], A has tracial rank zero. In Section 3 it was proved that A hasreal rank zero, stable rank one and has weakly unperforated K0(A), and so it suffices to showthe following: For any ε > 0 and a finite subset F ⊂ C(X ×T), there exists a projection e ∈ Ax

such that the following conditions hold.

• ∥ae − ea∥ < ε for all a ∈ F ∪ {u}.

• For any a ∈ F ∪ {u}, there exists b ∈ eAxe such that ∥eae − b∥ < ε.

21

• τ(1 − e) < ε for all τ ∈ T (A).

We may assume F∗ = F . Choose N ∈ N so that 2π/N is less than ε. Applying Lemma 5.5to N , ε/2 and a finite subset

G =N−1⋃

i=0

uiFui∗,

we obtain M > N , a clopen neighborhood U of x and a partial isometry w ∈ Ax. Put p = 1U

and q = 1αM (U). For t ∈ [0, π] we define

P (t) = p cos2 t + w sin t cos t + w∗ sin t cos t + q sin2 t.

Then P (t) is a continuous path of projections with P (0) = p and P (π) = q. By the choice of wwe obtain the estimate

∥ui∗P (t)uif − fui∗P (t)ui∥ < ε

for all t ∈ [0, π], i = 0, 1, . . . , N − 1 and f ∈ F . We define a projection e by

e = 1 −

(M−N∑

i=0

uipui∗ +N−1∑

i=1

ui∗P (iπ/N)ui

).

The partial isometry w satisfies ui∗wui ∈ Ax for all i = 1, 2, . . . , N − 1, and so e is a projectionof Ax. Evidently we have

∥fe − ef∥ < ε

for all f ∈ F . Since

∥P (iπ/N) − P ((i − 1)π/N)∥ <2π

N< ε,

it is not hard to see∥ue − eu∥ < ε.

It is clear that efe belongs to Ax for all f ∈ C(X × T). It follows from eue = eu(1 − p)e thateue also belongs to Ax. We can easily verify

τ(1 − e) < Mτ(p) <ε

2

for all τ ∈ T (A).

6 The generalized Rieffel projection

By Lemma 2.4, the Ki-group (i = 1, 2) of the crossed product C∗-algebra arising from anorientation preserving homeomorphism α×ϕ is isomorphic to the direct sum of Z and K0(X, α).Needless to say, the equivalence class of the implementing unitary is the generator of Z in theK1-group. This section is devoted to specify a projection of C∗(X × T, α × ϕ) which gives arepresentative of the generator of the Z-summand of the K0-group.

At first, we consider the case that a cocycle takes its values in the rotation group of thecircle. Let (X,α) be a Cantor minimal system and let ξ : X → T be a continuous map. Wedenote C∗(X × T, α × Rξ) by A for short and denote the implementing unitary by u. We willidentify K0(X, α) as a subgroup of K0(A).

Let ξ ∈ C(X, R) be an arbitrary lift of ξ ∈ C(X, T). Then Mα ∋ µ 7→ µ(ξ) ∈ R gives anaffine function on the set of invariant probability measures Mα. The other lifts of ξ are of theform ξ+f with f ∈ C(X, Z), and so this affine function is uniquely determined up to the natural

22

image of K0(X, α) in Aff(Mα). Suppose [ξ] = 0 in K0T(X, α). Then there exists η ∈ C(X, T)

such that ξ = η− ηα−1. When η ∈ C(X, R) is a lift of η, ξ = η− ηα−1 is a lift of ξ and µ(ξ) = 0for all µ ∈ Mα. Therefore we obtain a homomorphism

∆ : K0T(X, α) ∋ [ξ] 7→ ∆([ξ]) ∈ Aff(Mα)/D(K0(X, α)).

This homomorphism ∆ is known to be surjective ([M1, Lemma 6.2]).

Lemma 6.1. Let (X, α) be a Cantor minimal system and let ξ : X → T be a continuous map.For any ε > 0, there exists η ∈ C(X, T) such that |(ξ + η − ηα)(x)| < ε for all x ∈ X.

Proof. LetP = {X(v, k) : v ∈ V, k = 1, 2, . . . , h(v)}

be a Kakutani-Rohlin partition of (X, α) such that h(v) > ε−1 for every v ∈ V . We denote theroof set

⋃v∈V X(v, h(v)) by R(P). Put

κ(x) =h(v)∑

k=1

ξ(αk−1(x))

for all x ∈ X(v, 1) and v ∈ V . Since X is totally disconnected, there exists a real valuedcontinuous function κ on α(R(P)) such that

κ(x) + Z = κ(x) and − 1 < κ(x) < 1

for all x ∈ α(R(P)) =⋃

v∈V X(v, 1). Define η ∈ C(X, T) by η(x) = 0 for all x ∈ α(R(P)) and

η(αk(x)) =k∑

i=1

ξ(αi−1(x)) − k

h(v)κ(x) + Z

for x ∈ X(v, 1), v ∈ V and k = 1, 2, . . . , h(v) − 1. It is not hard to see that η is the desiredfunction.

In a similar fashion to the lemma above, we can show the following, which will be used later.

Lemma 6.2. Let (X,α) be a Cantor minimal system and let ξ : X → T and c : X → Z2 becontinuous maps. For any ε > 0, there exists η ∈ C(X, T) such that

|ξ(x) + η(x) − (−1)c(x)ηα(x)| < ε

for all x ∈ X.

Definition 6.3. Let (X,α) be a Cantor minimal system and let ξ : X → T be a continuousmap. Define

H(α, ξ) = {η ∈ C(X, T) : (ξ + η − ηα)(x) ∈ (1/10, 9/10) for all x ∈ X}.

By Lemma 6.1, H(α, ξ) is not empty.

Suppose that η belongs to H(α, ξ). We define a projection e(α, ξ, η) in A as follows. Definea real valued continuous function gη on X × T by

gη(x, t) =

{√10(t − η(x))(1 − 10(t − η(x))) t ∈ [η(x), η(x) + 1/10]

0 otherwise.

23

Put η′ = (ξ + η) ◦ α−1 and define a real valued continuous function f(α, ξ, η) on X × T by

f(α, ξ, η)(x, t) =

10(t − η(x)) t ∈ [η(x), η(x) + 1/10]1 t ∈ [η(x) + 1/10, η′(x)]1 − 10(t − η′(x)) t ∈ [η′(x), η′(x) + 1/10]0 otherwise.

Then it is easy to check that e(α, ξ, η) = gηu∗ + f(α, ξ, η) + ugη ∈ A is a self-adjoint projection.

We call e(α, ξ, η) the generalized Rieffel projection. Let ξ ∈ C(X, R) be a lift of ξ + η − ηα suchthat 1/10 < ξ < 9/10. Then, for every µ ∈ Mα,

τµ(e(α, ξ, η)) = τµ(f(α, ξ, η)) = µ(ξ),

where τµ is the tracial state on A corresponding to µ. Hence the affine function µ 7→ τµ(e(α, ξ, η))is a representative of ∆([ξ]) ∈ Aff(Mα)/D(K0(X,α)).

Proposition 6.4. In the situation above, let e = e(α, ξ, η) ∈ A be the generalized Rieffel pro-jection. Then K0(X,α) and [e] generate K0(A).

Proof. Let v the unilateral shift on ℓ2(N) and let T be the Toeplitz algebra generated by v.Put q = 1 − vv∗. In the C∗-algebra A ⊗ T , we consider the C∗-subalgebra B generated byC(X × T) ⊗ 1 and u ⊗ v∗. There is a surjective homomorphism π from B to A sending u ⊗ v∗

to u. The kernel of π is C(X × T) ⊗ K, where K is the algebra of compact operators. Puta = gηu

∗ ⊗ v + f(α, ξ, η) ⊗ 1 + ugη ⊗ v∗. Then π(a) = e. Hence it suffices to show that e2π√−1a

is a generator of K1(C1X ⊗ C(T)) ∼= Z. Define a continuous function h(t) by

h(t) = (1 − 8t(1 − t)) +√−1t(t − 1)(t − 1/2).

Since e2π√−1t is homotopic to h(t) in the set of complex-valued invertible functions, it suffices

to know the K1-class of the invertible element h(a). By

a − a2 = g2η ⊗ q

and(a2 − a)(a − 1/2) = −(g2

η(f(α, ξ, η) − 1/2)) ⊗ q,

it follows that h(a) is homotopic to 1X ⊗ z ⊗ q in GL(C(X × T)) ⊗ q, where z(t) = e2π√−1t is

the generator of K1(C(T)).

Let K0(A) ∼= Z⊕K0(X,α) be the isomorphism described in the proposition above. If α×Rξ

is minimal, then it follows from Theorem 3.13 that

K0(A)+ ∼= {(n, [f ]) : µ(nξ + f) > 0 for all µ ∈ Mα} ∪ {0}.

See also Remark 4.6.

We now turn to the general case. Let (X,α) be a Cantor minimal system and let ϕ :X → Homeo+(T) be a continuous map. We write the crossed product C∗-algebra arising from(X×T, α×ϕ) by A and the implementing unitary by u. In order to define the Rieffel projectionin A, we need some preparations. For ϕ ∈ Homeo+(T), let r(ϕ) ∈ T denote the rotation number.The reader may refer to [KH, Chapter 11] for the definition and some elementary properties ofr(ϕ).

24

Lemma 6.5. Let ϕ ∈ Homeo+(T). The map T ∋ t 7→ r(Rtϕ) is a continuous surjection from Tto T of degree one.

Proof. By the definition of the rotation number, we see that the mapping t 7→ r(Rtϕ) is nonde-creasing as a real valued function. It is clear that r(Rtϕ) = 0 if and only if t belongs to

I = {s − ϕ(s) : s ∈ T}.

Since ϕ is an orientation preserving homeomorphism, I is not the whole circle. Thus, I is aclosed interval of the circle. It follows that the map is a surjection of degree one.

Lemma 6.6. Let X be the Cantor set and let I ⊂ T be an open subset. When Φ : X × T → Tis a continuous map and Φ(x, ·) is surjective for every x ∈ X, there exists a continuous mapξ : X → T such that Φ(x, ξ(x)) ∈ I for all x ∈ X.

Proof. By assumption, for each x ∈ X, there exists tx ∈ T such that Φ(x, tx) ∈ I. The continuityof Φ implies that there exists a clopen neighborhood Ux of x such that Φ(Ux, tx) ⊂ I. Since X iscompact, it is covered by finitely many Ux’s. We can find a locally constant function ξ ∈ C(X, T)satisfying the required property.

The following lemma corresponds to Lemma 6.1.

Lemma 6.7. Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo+(T) be a con-tinuous map. Suppose that an open subset I ⊂ T is given. Then there exists η ∈ C(X, T) suchthat

r(Rηα(x) ◦ ϕx ◦ R−1η(x)) ∈ I

for all x ∈ X.

Proof. Define Φ(x, t) = r(Rtϕx). It is obvious that Φ is continuous. By Lemma 6.5, for each x ∈X, Φ(x, ·) is a continuous surjection. It follows from Lemma 6.6 that there exists a continuousmap ξ : X → T such that Φ(x, ξ(x)) ∈ I for all x ∈ X. Moreover, there exists ε > 0 such that,for all x ∈ X, if |t− ξ(x)| < ε then Φ(x, t) ∈ I. From Lemma 6.1, we can find η ∈ C(X, T) suchthat

|(ξ + η − ηα)(x)| < ε

for all x ∈ X. Then we get

r(Rηα(x) ◦ ϕx ◦ R−1η(x))

= r(Rη(x) ◦ Rηα(x)−η(x) ◦ ϕx ◦ R−η(x))

= r(Rηα(x)−η(x) ◦ ϕx)

= Φ(x, ηα(x) − η(x)) ∈ I,

thereby completing the proof.

By the lemma above, without loss of generality, we may always assume that r(ϕx) is notzero for all x ∈ X. Put

c = inf{|ϕx(t) − t|, |ϕ−1x (t) − t| : (x, t) ∈ X × T}.

Since ϕx has no fixed points, c is a positive real number. Take s ∈ T. We define a functionf(α, ϕ, s) ∈ C(X × T) by

f(α, ϕ, s)(x, t) =

c−1(t − s) t ∈ [s, s + c]1 t ∈ [s + c, ϕα−1(x)(s)]1 − c−1(ϕ−1

α−1(x)(t) − s) t ∈ [ϕα−1(x)(s), ϕα−1(x)(s + c)]

0 otherwise.

25

By the choice of c, f is well-defined. Define a function gs ∈ C(X × T) by

gs(x, t) =

{√c−1(t − s)(1 − c−1(t − s)) t ∈ [s, s + c]

0 otherwise.

Then one checks that e(α, ϕ, s) = gsu∗ + f(α, ϕ, s) + ugs is a well-defined projection of A. Let

us call it the generalized Rieffel projection for A. In exactly the same way as Proposition 6.4,we can show the following.

Proposition 6.8. In the above setting, K0(A) is generated by K0(X, α) and [e(α, ϕ, s)]. Fur-thermore, T ∋ s 7→ e(α,ϕ, s) ∈ A is a continuous path of projections in A.

In the definition of e(α,ϕ, s), we can replace gs by zgs, where z is a complex number with|z| = 1. But, this choice does not matter to the homotopy equivalence class of the projection.

7 Approximate K-conjugacy

Let us begin with recalling the definition of weakly approximate conjugacy.

Definition 7.1 ([LM, Definition 3.1]). Let (X,α) and (Y, β) be dynamical systems on com-pact metrizable spaces X and Y . We say that (X, α) and (Y, β) are weakly approximatelyconjugate, if there exist homeomorphisms σn : X → Y and τn : Y → X such that σnασ−1

n

converges to β in Homeo(Y ) and τnβτ−1n converges to α in Homeo(X).

Here we use the topology on Homeo(X) defined in Section 2. In other words, in the abovedefinition, we require that

limn→∞

∥g ◦ σnα−1σ−1n − g ◦ β−1∥ = 0

andlim

n→∞∥f ◦ τnβ−1τ−1

n − f ◦ α−1∥ = 0

for all g ∈ C(Y ) and f ∈ C(X).In [LM, Theorem 4.13], it was shown that two Cantor minimal systems are weakly approx-

imately conjugate if and only if they have the same periodic spectrum (see also [M3, Theorem3.1]). Similar results were shown in [M3] for dynamical systems on the product of the Cantorset and the circle.

As one sees that in the above definition, there is no consistency among σn or τn. It is clear (see[LM]) that the relation can be made stronger if one requires some consistency among σn as wellas τn. We hope such stronger version of approximate conjugacy is more reasonable replacementof (flip) conjugacy.

Suppose σnασ−1n → β in Homeo(Y ). In [LM, Proposition 3.2], it was shown that there exists

an asymptotic morphism {ψn} : B → A such that

limn→∞

∥ψn(f) − f ◦ σn∥ = 0

for all f ∈ C(Y ) andlim

n→∞ψn(uβ) = uα,

where uα and uβ denote the implementing unitaries in C∗(X,α) and C∗(Y, β). This observation,however, is far from the existence theorem in classification theory, which requires that {ψn} car-ries an isomorphism of K-groups (see [L5, Theorem 4.3] for instance). As pointed out in [LM],

26

we have to impose conditions on the conjugating maps σn so that the associated asymptoticmorphism has a nice property. Taking account of this, we make the following definitions. Byan order and unit preserving homomorphism ρ : K∗(B) → K∗(A), we mean a pair of homomor-phisms ρi : Ki(A) → Ki(B) (i = 0, 1) such that ρ0([1A]) = [1B] and ρ0(K0(A)+) ⊂ K0(B)+.

Definition 7.2. Let (X,α) and (Y, β) be dynamical systems on compact metrizable spaces Xand Y . Suppose that a sequence of homeomorphisms σn : X → Y satisfies σnασ−1

n → β inHomeo(Y ). Let {ψn} be the asymptotic morphism arising from σn. We say that the sequence{σn} induces an order and unit preserving homomorphism ρ : K∗(C∗(Y, β)) → K∗(C∗(X,α)) be-tween K-groups, if for every projection p ∈ M∞(C∗(Y, β)) and every unitary u ∈ M∞(C∗(Y, β)),there exists N ∈ N such that

[ψn(p)] = ρ([p]) ∈ K0(C∗(X,α)) and [ψn(u)] = ρ([u]) ∈ K1(C∗(X, α))

for every n ≥ N .

Definition 7.3 ([L6, Definition 5.3]). Let (X, α) and (Y, β) be dynamical systems on compactmetrizable spaces X and Y . We say that (X, α) and (Y, β) are approximately K-conjugate, ifthere exist homeomorphisms σn : X → Y , τn : Y → X and an isomorphism ρ : K∗(C∗(Y, β)) →K∗(C∗(X,α)) between K-groups such that

σnασ−1n → β, τnβτ−1

n → α

and the associated asymptotic morphisms {ψn} : B → A and {ϕn} : A → B induce theisomorphisms ρ and ρ−1.

We say that (X, α) and (Y, β) are approximately flip K-conjugate, if (X, α) is approximatelyK-conjugate to either of (Y, β) and (Y, β−1).

J. Tomiyama [T] proved that two topological transitive systems (X, α) and (Y, β) are flipconjugate if and only if there is an isomorphism ϕ : C∗(X,α) → C∗(Y, β) such that ϕ◦jα = jβ ◦χfor some isomorphism χ : C(X) → C(Y ).

Definition 7.4 ([L6, Definition 3.8]). Let (X, α) and (Y, β) be two topological transitivesystems. We say that (X,α) and (Y, β) are C∗-strongly approximately flip conjugate if thereexists a sequence of isomorphisms ϕn : C∗(X, α) → C(Y, β) and a sequence of isomorphismsχn : C(X) → C(Y ) such that [ϕn] = [ϕ1] in KL(C∗(X,α), C∗(Y, β)) for all n ∈ N and

limn→∞

∥ϕn ◦ jα(f) − jβ ◦ χn(f)∥ = 0

for all f ∈ C(X).

Let X be the Cantor set and let ξ be a continuous function from X to T. In this section, wewould like to discuss approximate K-conjugacy for (X × T, α × Rξ).

As in the previous section, let A denote the crossed product C∗-algebra C∗(X ×T, α×Rξ).By Lemma 6.1, we can find a continuous function ζ : X → T such that [ζ] = [ξ] in K0

T(X, α) andζ(x) ∈ (7/15, 8/15) for every x ∈ X. By Lemma 4.3, α × Rξ is conjugate to α × Rζ . Thereforewe may assume ξ(x) ∈ (7/15, 8/15) for all x ∈ X without loss of generality.

Suppose that η belongs to H(α, ξ). Let η ∈ C(X, R) be a lift of η. Since (ξ + η − ηα)(x) ∈(1/10, 9/10) for every x ∈ X, we have (η − ηα)(x) /∈ [13/30, 17/30]. Hence there exists a uniquef ∈ C(X, Z) such that

∥f − (η − ηα)∥∞ <12.

It is easy to see that [f ] ∈ K0(X,α) is independent of the choice of η. Let us denote [f ] byBα(η).

27

Lemma 7.5. In the above setting, suppose that η and η′ are homotopic in H(α, ξ).

(1) e(α, ξ, η) is homotopic to e(α, ξ, η′) in the set of projections of A.

(2) Bα(η) = Bα(η′).

Proof. (1) Suppose [0, 1] ∋ t 7→ κt ∈ H(α, ξ) is a homotopy from η to η′. Then the general-ized Rieffel projection e(α, ξ, κt) is well-defined and t 7→ e(α, ξ, κt) gives a continuous path ofprojections in A from e(α, ξ, η) to e(α, ξ, η′).

(2) Let κt be a homotopy as in (1). There exists a continuous map κ from X × [0, 1] to Rsuch that

κt(x) + Z = κt(x)

for all x ∈ X and t ∈ [0, 1]. Since

(κt − κtα)(x) = 12,

the integer nearest to (κt − κtα)(x) does not vary as t varies. Hence we get Bα(η) = Bα(η′).

Let η ∈ H(α, ξ). The unitary vη(x) = e2π√−1η(x) of C(X) satisfies ∥uαvηu

∗αv∗η − 1∥∞ < 2,

where uα denotes the implementing unitary of C∗(X, α), because of (η−ηα)(x) = 1/2. Thus, uα

and vη are almost commuting unitaries in a sense. When u, v ∈ Mn(C) are unitaries satisfying∥uv − vu∥ < 2, on account of det(uvu∗v∗) = 1, we have

12π

√−1

Tr(log(uvu∗v∗)) ∈ Z ∼= K0(Mn(C)),

where Tr is the standard trace on Mn(C) and log is the logarithm with values in {z : ℑ(z) ∈(−π, π)}. The Bott element for pairs of almost commuting unitaries in a unital C∗-algebra is ageneralization of this (see [EL]). More precisely, if u and v are unitaries in a unital C∗-algebraand ∥uv−vu∥ ≈ 0, then a projection B(u, v) and an element of the K0-group are obtained. OurBα(η) is just this K0-class for uα and vη.

Lemma 7.6. Suppose (ξ + η − ηα)(x) ∈ (1/3, 2/3) for all x ∈ X. Then we have

[e(α, ξ, η)] = [e(α, ξ, 0)] − Bα(η)

in K0(A).

Proof. Put

c = inf{|(ξ + η − ηα)(x)| : x ∈ X} − 13.

Then c is positive. Choose a sufficiently finer Kakutani-Rohlin partition

P = {X(v, k) : v ∈ V, 1 ≤ k ≤ h(v)}

for (X, α). Let R(P) be the roof set of P. We may assume that h(v) is large and

sup{|η(x) − η(y)| : x, y ∈ U} <c

2

for every U ∈ P, where

P = {X(v, k) : v ∈ V, 1 ≤ k < h(v)} ∪ {R(P)}.

Take xU ∈ U for all U ∈ P and define η′ ∈ C(X, Z) by η′(x) = η(xU ) for x ∈ U . It is not hardto see that η and η′ are homotopic in H(α, ξ). By Lemma 7.5, [e(α, ξ, η)] = [e(α, ξ, η′)] and

28

Bα(η) = Bα(η′). Hence, by replacing η by η′, we may assume that η is constant on each clopenset belonging to P. Furthermore, by adding a constant function, we may assume that η(x) = 0for all x ∈ R(P).

Note that |(η − ηα)(x)| is less than 2/3 − 7/15 = 1/5 for all x ∈ X. There is a unique liftη ∈ C(X, R) of η such that η(x) = 0 for all x ∈ R(P) and |(η − ηα)(y)| < 1/5 for all y ∈ R(P)c.Moreover, for every v ∈ V , there exists an integer mv such that

|mv − η(x)| <15

for all x ∈ X(v, 1). Therefore Bα(η) is equal to∑

v∈V −mv[1X(v,1)], and

|mv| = |mv − η(αh(v)−1(x))| ≤ |mv − η(x)| +h(v)−1∑

k=1

|η(αk−1(x)) − η(αk(x))| <h(v)

5,

where x is a point in X(v, 1).Fix v0 ∈ V . Let a : {1, 2, . . . , h(v0)} → R be a map such that a(h(v0)) = 0, |mv0 − a(1)| <

11/30 and |a(k) − a(k + 1)| < 11/30 for all k = 1, 2, . . . , h(v0) − 1. Define a continuous map κfrom X to R by

κ(x) =

{a(k) x ∈ X(v0, k), k = 1, 2, . . . , h(v0)η(x) otherwise.

Put κ(x) = κ(x) + Z. Then κ ∈ C(X, T) belongs to H(α, ξ), because 7/15 − 11/30 = 1/10. Fort ∈ [0, 1], put κt = tκ + (1 − t)η. Then it is not hard to see that κt gives a homotopy from η toκ in H(α, ξ).

At first, let us consider the case that mv0 is positive. Define a : {1, 2, . . . , h(v0)} → R bya(1) = a(2) = mv0 + 1

3 , a(3) = mv0 and

a(k) =mv0(h(v0) − k)

h(v0) − 3

for every k = 4, 5, . . . , h(v0). By using this map a, define κ ∈ C(X, R) be as above. It followsthat η and κ are homotopic in H(α, ξ). Let f1 be the continuous function on X × T defined by

f1(x, t) =

10(t − 1/3) (x, t) ∈ X(v0, 2) × [1/3, 1/3 + 1/10]1 (x, t) ∈ X(v0, 2) × [1/3 + 1/10, 2/3]1 − 10(t − 2/3) (x, t) ∈ X(v0, 2) × [2/3, 2/3 + 1/10]0 otherwise.

Let U = X(v0, 2) × T and put f2 = (1U − f1) ◦ (α × Rξ)−1. Define a continuous function g onX × T by

g(x, t) =

−√

10(t − 1/3)(1 − 10(t − 1/3)) (x, t) ∈ X(v0, 2)× ∈ [1/3, 1/3 + 1/10]√10(t − 2/3)(1 − 10(t − 2/3)) (x, t) ∈ X(v0, 2)× ∈ [2/3, 2/3 + 1/10]

0 otherwise.

Then it can be verified that e = gu∗ + (f1 + f2) + ug is a projection and e is equivalent to 1U :in fact, if h ∈ C(X × T) is a function with

h|X(v0, 2) × [1/3, 1/3 + 1/10] = −1, h|X(v0, 2) × [2/3, 2/3 + 1/10] = 1

29

and |h|2 = 1, then the partial isometry

w = h√

f1 +√

1U − f1u∗

satisfies w∗w = e and ww∗ = 1U . Furthermore e is a subprojection of e(α, ξ, κ) and

e(α, ξ, κ) − e = e(α, ξ, η′)

for some η′ ∈ H(α, ξ) with Bα(η′) = Bα(η) + [1X(v0,1)]. Hence

[e(α, ξ, 0)] − [e(α, ξ, η)] − Bα(η) = [e(α, ξ, 0)] − [e(α, ξ, κ)] − Bα(η)= [e(α, ξ, 0)] − [e(α, ξ, η′)] − [e] − Bα(η)= [e(α, ξ, 0)] − [e(α, ξ, η′)] − Bα(η′).

We can repeat the same argument with η′ in place of η. By repeating this mv0 times, we willobtain

[e(α, ξ, 0)] − [e(α, ξ, η)] − Bα(η) = [e(α, ξ, 0)] − [e(α, ξ, η′)] − Bα(η′)

with Bα(η′) =∑

v =v0−mv[1X(v,1)] and η′(x) = 0 for all x ∈ X(v0, 1) ∪ · · · ∪ X(v0, h(v0)).

When mv0 is negative, there exists κ homotopic to η in H(α, ξ) such that e(α, ξ, κ) + e =e(α, ξ, η′) for some η′ ∈ H(α, ξ) with Bα(η′) = Bα(η) − [1X(v0,1)]. In a similar fashion to thepreceding paragraph, the same conclusion follows.

By applying the same argument to all towers in V , we have

[e(α, ξ, 0)] − [e(α, ξ, η)] − Bα(η) = 0.

Lemma 7.7. Let (X,α) be a Cantor minimal system. Suppose that ξ1, ξ2 ∈ C(X, R) andf ∈ C(X, Z) satisfy

µ(ξ2) = µ(ξ1) + µ(f)

for every α-invariant measure µ ∈ Mα and

715

< ξi(x) <815

for all x ∈ X and i = 1, 2. Put ξi(x) = ξi(x) + Z for i = 1, 2. Then, for any ε > 0, there existsη ∈ C(X, T) such that

|(ξ1 − ξ2)(x) − (η − ηα)(x)| < ε

for all x ∈ X and Bα(η) = [f ].

Proof. We may assume ε < 10−1. In the same way as in [GW, Lemma 2.4], we can find aKakutani-Rohlin partition

P = {X(v, k) : v ∈ V, k = 1, 2, . . . , h(v)}

such that1

h(v)

∣∣∣∣∣∣

h(v)∑

k=1

(ξ1 − ξ2 + f)(αk−1(x))

∣∣∣∣∣∣< ε

for all v ∈ V and x ∈ X(v, 1). Define a real valued continuous function κ on α(R(P)) by

κ(x) =h(v)∑

k=1

(ξ1 − ξ2 + f)(αk−1(x))

30

for x ∈ X(v, 1). Let η be a real valued continuous function on X such that

η(αk(x)) =k∑

i=1

(ξ2 − ξ1)(αi−1(x)) +k

h(v)κ(x)

for all x ∈ X(v, 1) and k = 0, 1, . . . , h(v) − 1. Define η(x) = η(x) + Z. It is straightforward tocheck

|(ξ1 − ξ2)(x) − (η − ηα)(x)| < ε

for all x ∈ X. For x /∈ R(P), we have

|(η − ηα)(x)| <115

+110

,

and so the integer nearest to (η − ηα)(x) is zero. If x belongs to the roof set R(P), then

|(η − ηα)(x) −h(v)∑

k=1

f(α1−k(x))| <115

+110

.

Hence we can conclude that Bα(η) is equal to [f ].

Now we are ready to prove the main theorem of this section. Let (X,α) and (Y, β) be Cantorminimal systems and let ξ : X → T and ζ : Y → T be continuous functions. Suppose that α×Rξ

and β×Rζ are both minimal. We denote by A (resp. B) the crossed product C∗-algebra arisingfrom (X × T, α × Rξ) (resp. (Y × T, β × Rζ)).

Theorem 7.8. The following are equivalent.

(1) (X × T, α × Rξ) is approximately K-conjugate to (Y × T, β × Rζ).

(2) There exists a unital order isomorphism ρ from K0(B) to K0(A) such that ρ(K0(Y, β)) =K0(X,α).

Proof. (1)⇒(2). This is immediate from the definition of approximate K-conjugacy (Definition7.3).

(2)⇒(1). By Lemma 6.1, without loss of generality, we may assume ξ(x), ζ(y) ∈ (7/15, 8/15)for all x ∈ X and y ∈ Y . Let e(α, ξ, 0) ∈ A and e(β, ζ, 0) ∈ B be the projections described inthe previous section. We follow the notation used there. Since ρ is an isomorphism, there areonly two possibilities:

ρ([e(β, ζ, 0)]) − [e(α, ξ, 0)] ∈ K0(X,α)

orρ([e(β, ζ, 0)]) + [e(α, ξ, 0)] ∈ K0(X, α).

Suppose that the latter equality holds. The dynamical system (X × T, α × Rξ) is conjugate to(X × T, α × R−ξ) via the mapping (x, t) 7→ (x,−t). This conjugacy induces an isomorphismbetween the corresponding C∗-algebras, which in turn yields an isomorphism between the K0-groups. One can see that [e(α, ξ, 0)] is sent to [1 − e(α,−ξ, 0)] by this isomorphism. For thisreason, by replacing α × Rξ by α × R−ξ, we may always assume that there exists h ∈ C(X, Z)such that

ρ([e(β, ζ, 0)]) = [e(α, ξ, 0)] + [h]α

in K0(A).

31

The restriction of ρ to K0(Y, β) is a unital order isomorphism onto K0(X, α). By [LM,Theorem 5.4] or [M3, Theorem 3.4], there exist homeomorphisms σn : X → Y such that

σnασ−1n → β

in Homeo(Y ) and[fσn]α = ρ([f ]β)

for all f ∈ C(Y, Z) and n ∈ N. We may assume

|ζ ◦ β−1σn(x) − ζ ◦ σnα−1(x)| <1n

for all x ∈ X and n ∈ N. As in Remark 4.6, we denote the state on the K0-group arising froman invariant measure µ by Sµ. Then, for any µ ∈ Mα and n ∈ N, we have

ρ∗(Sµ)([f ]β) = Sµ(ρ([f ]β)) = µ([fσn]α) = Sσn∗(µ)([f ]β)

for all f ∈ C(Y, Z). Thus, ρ∗(Sµ) = Sσn∗(µ) on K0(Y, β), and so on K0(B) (see Remark 4.6).Let ξ ∈ C(X, R) and ζ ∈ C(Y, R) be the lifts of ξ and ζ satisfying

715

< ξ(x) <815

and715

< ζ(y) <815

for all x ∈ X and y ∈ Y . It follows that

µ(ζσn) = σn∗(µ)(ζ) = Sσn∗(µ)([e(β, ζ, 0)])

= ρ∗(Sµ)([e(β, ζ, 0)]) = Sµ(ρ([e(β, ζ, 0)])= Sµ([e(α, ξ, 0)] + [h]α)

= µ(ξ) + µ(h)

for every µ ∈ Mα and n ∈ N. Now Lemma 7.7 applies and yields ηn ∈ C(X, T) such that

|(ξ − ζσn)(x) − (ηn − ηnα)(x)| <1n

for all x ∈ X and Bα(ηn) = [h]. Hence it is easy to verify that

(σn × Rηn)(α × Rξ)(σn × Rηn)−1 → (β × Rζ)

in Homeo(Y × T). Furthermore, since

|ζβ−1σn(x) − (ξα−1 + ηn − ηnα−1)(x)| <2n

for all x ∈ X, we get the estimate

∥f(β, ζ, 0) ◦ (σn × Rηn) − f(α, ξ,−ηn)∥ <20n

.

See the discussion before Proposition 6.4 for the definition of f(·, ·, ·). It is easy to see g0 ◦ (σn ×Rηn) = g−ηn . Therefore, the asymptotic morphism {ψn} : B → A associated with σn × Rηn

satisfieslim

n→∞∥ψn(e(β, ζ, 0)) − e(α, ξ,−ηn)∥ = 0,

32

and Lemma 7.6 yields

[e(α, ξ,−ηn)] = [e(α, ξ, 0)] + Bα(ηn)= [e(α, ξ, 0)] + [h]α = ρ([e(β, ζ, 0)]).

For every clopen set U ⊂ Y , we know

limn→∞

∥ψn(1U ) − 1U ◦ σn∥ = 0

and [1U ◦ σn]α = ρ([1U ]β). It follows that {ψn} induces ρ : K0(B) → K0(A).For every clopen set U ⊂ Y , we know

limn→∞

∥ψn(z1U ) − z1U ◦ (σn × Rηn)∥ = 0,

where z is a unitary defined by z(y, t) = e2π√−1t. It is clear that

[1Uc + z1U ◦ (σn × Rηn)] = [1Uc + z(1U ◦ σn)]

in K1(A). Since {ψn} approximately carries the implementing unitary of β × Rζ to that ofα × Rξ, we can conclude that {ψn} induces an isomorphism between K1(B) and K1(A).

Consequently {ψn} induces an isomorphism between K-groups. Similarly, we can constructan asymptotic morphism from A to B which induces ρ−1 between their K-groups.

Theorem 7.9. Suppose that α×Rξ and β ×Rζ are minimal and rigid. Then the following areequivalent.

(1) (X × T, α × Rξ) is approximately K-conjugate to (Y × T, β × Rζ).

(2) There exists a unital order isomorphism ρ from K0(B) to K0(A) such that ρ(K0(Y, β)) =K0(X,α).

(3) (X × T, α × Rξ) is approximately flip K-conjugate to (Y × T, β × Rζ).

(4) α × Rξ and β × Rζ are C∗-strongly approximately flip conjugate.

(5) There is θ ∈ KL(A,B) which gives an order and unit preserving isomorphism from(K0(A),K0(A)+, [1A],K1(A)) onto (K0(B),K0(B)+, [1B], K1(B)) and an isomorphism χ :C(X × T) → C(Y × T) such that

[jα×Rξ] × θ = [jβ×Rζ

◦ χ]

in KL(C(X × T), B).

Proof. We have seen (1)⇔(2) in Theorem 7.8.(1)⇒(3) is obvious.When both α × Rξ and β × Rζ are rigid, A and B have tracial rank zero by Theorem 5.6.

Thus,(3)⇒(4) follows from [L6, Theorem 5.4].(4)⇒(5) follows immediately from [L6, Theorem 3.9].(5)⇒(2). Suppose that θ induces an order and unit preserving isomorphism Γ(θ) from

(K0(A),K0(A)+, [1A],K1(A)) to (K0(B),K0(B)+, [1B],K1(B)). Suppose that there is χ : C(X×T) → C(Y ×T) such that [jα×Rξ

]× θ = [jβ×Rζ◦χ]. This implies that Γ(θ) gives an isomorphism

from K0(X, α) onto K0(Y, β). So (2) holds.

33

8 Non-orientation preserving case

Let (X, α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuous map. Inthis section, we would like to consider the case that α×ϕ is not orientation preserving, that is,[o(ϕ)] is not zero in K0(X,α)/2K0(X, α). As was seen in the discussion before Lemma 2.5, theskew product extension (X × Z2, α × o(ϕ)) is a Cantor minimal system. This system will playan important role when we study α × ϕ.

Define a continuous map ϕ : X × Z2 → Homeo(T)+ by

ϕ(x,k) = λk+o(ϕ)(x)ϕxλk,

where λ is given by λ(t) = −t for t ∈ T. Let π be the projection from X × Z2 to the firstcoordinate.

Lemma 8.1. As a Homeo(T)-valued cocycle on the Cantor minimal system (X ×Z2, α× o(ϕ)),ϕπ is cohomologous to ϕ. In particular, ϕπ is orientation preserving with respect to the minimalhomeomorphism α × o(ϕ).

Proof. Put ω(x.k) = λk. Then

ϕ(x,k) ◦ ω(x,k) = λk+o(ϕ)(x) ◦ ϕx = ω(α×o(ϕ))(x,k) ◦ ϕπ(x,k)

implies that they are cohomologous.

We remark that the following diagram of factor maps is commutative:

(X × T, α × ϕ) π×id←−−−− (X × Z2 × T, α × o(ϕ) × ϕπ)y

y(X, α) ←−−−−

π(X × Z2, α × o(ϕ)).

From Lemma 6.5 and 6.6, there exists ξ ∈ C(X, T) such that r(Rξ(x)λo(ϕ)(x)ϕx) = 0 for all

x ∈ X. By applying Lemma 6.2 to −ξ and o(ϕ), we obtain η ∈ C(X, T) such that

|η(x) − (−1)o(ϕ)(x)ηα(x) − ξ(x)| < ε,

where ε is sufficiently small so that it implies

0 = r(Rη(x)−(−1)o(ϕ)(x)ηα(x)λo(ϕ)(x)ϕx)

= r(R−1(−1)o(ϕ)(x)ηα(x)

λo(ϕ)(x)ϕxRη(x))

= r(λo(ϕ)(x)R−1ηα(x)ϕxRη(x))

for all x ∈ X. Therefore, by perturbing ϕ by Rη, we may assume

r(ϕ(x,k)) = (−1)kr(λo(ϕ)(x)ϕx) = 0

for all (x, k) ∈ X × Z2.Let A denote the crossed product C∗-algebra arising from (X × T, α × ϕ). We write the

implementing unitary by u. Define an automorphism θ ∈ Aut(A) of order two by

θ(f) = f

for all f ∈ C(X × T) andθ(u) = ug,

where g ∈ C(X × T) is given by g(x, t) = (−1)o(ϕ)(x). By Lemma 6.1, it can be seen that θ isapproximately inner, and so it induces the identity on the K-group.

34

Proposition 8.2. In the situation above, the crossed product C∗-algebra arising from the dy-namical system (X × Z2 × T, α × o(ϕ) × ϕπ) is isomorphic to A oθ Z2.

Proof. We write the implementing unitary in A by uA. Let us denote C∗(X×Z2×T, α×o(ϕ)×ϕπ)by B and the implementing unitary in B by uB. Let v denote the unitary which implements θ.

We would like to define a homomorphism Φ from A o Z2 to B. The C∗-algebra A can benaturally embedded into B via the factor map π × id from X × Z2 × T to X × T. Let Φ|A bethis embedding. Define a continuous function h ∈ C(X × Z2 × T) by

h(x, k, t) = (−1)k.

Then,h ◦ (α × o(ϕ) × ϕπ) = h(g ◦ (π × id)).

Put Φ(v) = h. For f ∈ C(X × T), we have

hΦ(f)h = h(f ◦ (π × id))h = f ◦ (π × id) = Φ(θ(f)).

Besides,hΦ(uA)h = huBh = uB(g ◦ (π × id)) = Φ(θ(uA)).

It follows that Φ is a well-defined homomorphism. It is not hard to see that Φ is an isomorphism.

We freely use the identification of the two C∗-algebras established in the proposition above.By Lemma 8.1 and 2.4, we know that

K0(A o Z2) ∼= Z ⊕ K0(X × Z2, α × o(ϕ))

andK1(A o Z2) ∼= Z ⊕ K0(X × Z2, α × o(ϕ)).

On the crossed product C∗(X × Z2 × T, α × o(ϕ) × ϕπ), the dual action θ is given by

θ(f)(x, k, t) = f(x, k + 1, t)

for f ∈ C(X × Z2 × T) andθ(u) = u,

where u is the implementing unitary. Define a homeomorphism γ ∈ Homeo(X × Z2) by

γ(x, k) = (x, k + 1).

Then γ × id commutes with α × o(ϕ) × ϕπ and θ on C(X × Z2 × T) is induced by γ × id. Letus consider the induced action θ∗ on the K-groups. Evidently, θ∗0 on K0(X × Z2, α × o(ϕ)) isgiven by

[f ] 7→ [fγ],

and θ∗1 on K0(X × Z2, α × o(ϕ)) is given by

[f ] 7→ [−fγ].

Of course, θ∗([u]) = [u]. Hence it remains to know the image of the generalized Rieffel projection.Let

e = e(α × o(ϕ), ϕ, 0) = g0u∗ + f(α × o(ϕ), ϕ, 0) + ug0

35

be the projection of C∗(X × Z2 × T, α × o(ϕ) × ϕ) as in Proposition 6.8. This is well-defined,because r(ϕ(x,k)) is not zero. By the map

(x, k, t) 7→ (x, k + 1, λ(t)),

f(α × o(ϕ), ϕ, 0) is carried to a function supported on{(x, k, t) : t ∈ [−ϕ(α×o(ϕ))−1(x,k+1)(c), 0]

}

and g0 is carried to a function supported on X × Z2 × [−c, 0]. Note that

− ϕ(α×o(ϕ))−1(x,k+1)(c)

= −λk+1ϕα−1(x)λk+1+o(ϕ)(α−1(x))(c)

= −λϕ(α×o(ϕ))−1(x,k)λ(c)

= ϕ(α×o(ϕ))−1(x,k)(−c).

Hence, under the identification of C∗(X×Z2×T, α×o(ϕ)×ϕ) with C∗(X×Z2×T, α×o(ϕ)×ϕπ),we have

θ(e) = θ(e(α × o(ϕ), ϕ, 0))= g−cu

∗ + 1 − f(α × o(ϕ), ϕ,−c) + ug−c

= 1 − (−g−cu∗ + f(α × o(ϕ), ϕ,−c) − ug−c).

By the remark following Proposition 6.8, this is homotopic to

1 − e(α × o(ϕ), ϕ,−c),

and moreover it is homotopic to1 − e(α × o(ϕ), ϕ, 0)

from Proposition 6.8. Thus, we get[θ(e)] = 1 − [e]

in K0(A o Z2). In particular, θ is not approximately inner.Next, we would like to consider the map between K-groups induced from the inclusion

ι : A ↪→ A o Z2. On the K0-group, that is clearly given by

ι∗0([f ]) = (0, [f ◦ π]) ∈ Z ⊕ K0(X × Z2, α × o(ϕ)) ∼= K0(A o Z2)

for [f ] ∈ K0(X, α) ∼= K0(A). On the K1-group, for [f ] ∈ Coker(id−α∗ϕ), we can see that

ι∗1([f ]) = (0, [δ(f)]) ∈ Z ⊕ K0(X × Z2, α × o(ϕ)) ∼= K1(A o Z2),

where δ(f)(x, k) = (−1)kf(x) for (x, k) ∈ X × Z2. Under the identification

Coker(id−α∗ϕ) ∼= K0(X × Z2, α × o(ϕ))/K0(X, α)

established in the discussion before Lemma 2.5, this map is also described by

ι∗1([f ] + K0(X, α)) = (0, [f − fγ])

for f ∈ C(X × Z2, Z).

We now consider minimality and rigidity of α × ϕ.

36

Lemma 8.3. Let (X,α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuousmap. Suppose that α × ϕ is not orientation preserving. Then α × ϕ is minimal if and only ifα × o(ϕ) × ϕ is minimal.

Proof. We follow the notation used in the discussion above. By Lemma 8.1, we may replaceα × o(ϕ) × ϕ by α × o(ϕ) × ϕπ.

Suppose that α × o(ϕ) × ϕπ is minimal. If E ⊂ X × T is a closed α × ϕ-invariant subset,then (π × id)−1(E) is a closed α × o(ϕ) × ϕπ-invariant subset of X × Z2 × T. It follows that(π × id)−1(E) is empty or the whole X × Z2 × T. Namely E is empty or the whole X × T.

Let us prove the converse. Assume that α×ϕ is minimal. Let E ⊂ X ×Z2×T be a minimalsubset of α×o(ϕ)×ϕπ. Since (π× id)(E) is a closed α×ϕ-invariant subset, it must be equal toX ×T. If E = γ(E), then E = X ×Z2×T and we have nothing to do. Suppose that E∩γ(E) isempty and E ∪ γ(E) = X ×Z2 ×T. Thus E is a clopen subset. Hence there exists a continuousfunction χ : X → Z2 such that

E = {(x, χ(x), t) : x ∈ X, t ∈ T}.

It follows that χ + o(ϕ) = χα, which contradicts [o(ϕ)] = 0 in K0(X, α)/2K0(X,α).

Lemma 8.4. Let (X,α) be a Cantor minimal system and let ϕ : X → Homeo(T) be a continuousmap. Suppose that α × ϕ is not orientation preserving.

(1) If α × o(ϕ) × ϕ is rigid, then α × ϕ is rigid.

(2) If ϕ takes its values in Isom(T) and α × ϕ is rigid, then α × o(ϕ) × ϕ is rigid.

Proof. We follow the notation used in the discussion above. By Lemma 8.1, we may replaceα×o(ϕ)×ϕ by α×o(ϕ)×ϕπ. Let F (resp. F ) denote the canonical factor map from (X×T, α×ϕ)to (X, α) (resp. from (X × Z2 × T, α × o(ϕ) × ϕπ) to (X × Z2, α × o(ϕ)).

(1) Suppose that there exist two distinct ergodic measures ν1 and ν2 for (X ×T, α×ϕ) suchthat F∗(ν1) = F∗(ν2). Let νi be an α×o(ϕ)×ϕπ-invariant measure such that (π× id)∗(νi) = νi.Of course, ν1 = ν2, because of

(π × id)∗(ν1) = ν1 = ν2 = (π × id)∗(ν2).

By replacing νi by12(νi + (γ × id)∗(νi)),

we may assume that νi is γ × id-invariant. It follows that F∗(νi) is invariant under γ. Togetherwith

π∗F∗(ν1) = F∗(π × id)∗(ν1) = F∗(ν1) = F∗(ν2) = F∗(π × id)∗(ν2) = π∗F∗(ν2),

we have F∗(ν1) = F∗(ν2). Therefore α × o(ϕ) × ϕπ is not rigid.(2) Assume that α × o(ϕ) × ϕπ is not rigid. There exists an ergodic measure µ for (X ×

Z2, α × o(ϕ)) such that F−1∗ (µ) is not a singleton. By assumption, ϕ takes its values in the

rotation group. It follows from Lemma 4.4 and its proof that there exist uncountably manyergodic measures for (X × Z2 × T, α × o(ϕ) × ϕπ) in F−1

∗ (µ). In particular, we can find twodistinct ergodic measures ν1, ν2 ∈ F−1

∗ (µ) such that (γ×id)∗(ν1) = ν2. Hence, it is easily verifiedthat

(π × id)∗(ν1) = (π × id)∗(ν2)

in Mα×ϕ. But, we have

F∗(π × id)∗(ν1) = π∗F∗(ν1) = π∗(µ) = π∗F∗(ν2) = F∗(π × id)∗(ν2),

and so α × ϕ is not rigid.

37

Now we consider cocycles with values in Isom(T). Let (X, α) be a Cantor minimal system andlet ϕ : X → Isom(T) be a continuous map. There exists ξ ∈ C(X, T) such that ϕx = λo(ϕ)(x)Rξ(x)

for all x ∈ X. Suppose that α × ϕ is not orientation preserving and not minimal. By Lemma8.3, α × o(ϕ) × ϕ is not minimal, where ϕ is given by

ϕ(x,k) = λk+o(ϕ)(x)ϕxλk = λkRξ(x)λk =

{Rξ(x) k = 0R−ξ(x) k = 1.

It is convenient to introduce δ : C(X, T) → C(X × Z2, T) defined by δ(η)(x, k) = (−1)kη(x). Itfollows from Lemma 4.2 that there exist n ∈ N and ζ ∈ C(X × Z2, Z) such that

nδ(ξ) = ζ − ζ ◦ (α × o(ϕ))−1.

We also have

−nδ(ξ) = nδ(ξ) ◦ γ = (ζ − ζ ◦ (α × o(ϕ))−1) ◦ γ = ζγ − ζγ ◦ (α × o(ϕ))−1.

Since α × o(ϕ) is minimal on X × Z2, ζ + ζγ must equal a constant function. We can adjust ζby a constant function so that ζ + ζγ is equal to zero. Thus there exists η ∈ C(X, T) such that

nδ(ξ) = δ(η) − δ(η) ◦ (α × o(ϕ))−1.

Combining this with δ ◦ α∗ϕ = (α × o(ϕ))∗ ◦ δ, we obtain

nξ = η − α∗ϕ(η).

Lemma 8.5. Let (X, α) be a Cantor minimal system and let ϕx = λo(ϕ)(x)Rξ(x) be a cocyclewith values in Isom(T). If α×ϕ is not orientation preserving and not minimal, then there existn ∈ N and η ∈ C(X, T) such that

nξ = η − α∗ϕ(η).

Moreover, every minimal subset of α × ϕ is given by

Es = {(x, t) : nt = α∗ϕ(η)(x) + s or nt = α∗

ϕ(η)(x) − s}

for some s ∈ T.

Proof. The first part follows the discussion above. Let us consider the latter part. By Lemma4.2, every minimal set of α × o(ϕ) × ϕ is given by

{(x, k, t) : nt = (−1)k+o(ϕ)(α−1(x))ηα−1(x) + s}.

This closed set is carried to

Es = {(x, t) : nt = α∗ϕ(η)(x) + s or nt = α∗

ϕ(η)(x) − s}

by the factor map from X × Z2 × T to X × T. Therefore Es is a minimal set of α × ϕ.

38

9 Examples

Example 9.1. Let θ ∈ T be an irrational number and let ξ : T → T be a continuous map. Ahomeomorphism

γ : (s, t) 7→ (s + θ, t + ξ(s))

on T2 is called a Furstenberg transformation. In [OP], the crossed product C∗-algebra arisingfrom (T2, γ) is studied. We would like to replace the irrational rotation with a Cantor minimalsystem and construct an almost one to one extension of (T2, γ) as follows. Let ϕ be a Denjoyhomeomorphism on T with r(ϕ) = θ as in Remark 3.2. The homeomorphism ϕ has the uniqueinvariant nontrivial closed subset X. Let α be the restriction of ϕ to X. Then (X, α) is a Cantorminimal system and there exists an almost one to one factor map π from (X,α) to the irrationalrotation (T, Rθ) (see [PSS] for details). Both (X, α) and (T, Rθ) are uniquely ergodic. It is easyto see that π × id : X × T → T2 satisfies

(π × id) ◦ (α × Rξπ) = γ ◦ (π × id),

that is, π× id is a factor map. One can check that if γ is minimal, then α×Rξπ is also minimal.The factor map π induces a Borel isomorphism between X and S1. Hence α × Rξπ is rigid ifand only if γ is uniquely ergodic. There are some known criterions for unique ergodicity of γ.For example, it was proved in [F, Theorem 2.1] that if ξ is a Lipschitz function and its degree isnot zero then γ is uniquely ergodic. On the other hand it is known that there exist θ ∈ T andξ : T → T such that γ is minimal but not uniquely ergodic (see [F, p.585] for instance). In thiscase α × Rξπ is minimal but not rigid.

Now let us consider the example of Putnam which was presented by N. C. Phillips in [Ph3].

Example 9.2. For any θ ∈ R \ Q let gθ be a minimal homeomorphism of a Cantor set Xθ ⊂ Tobtained from a Denjoy homeomorphism g0

θ : T → T as in [Ph3]. Choose g0θ to have rotation

number θ and such that the unique minimal set Xθ ⊂ T has the property that the image ofT \ Xθ under the semiconjugation to Rθ is a single orbit of Rθ.

Now let θ1, θ2 ∈ R \ Q be irrational numbers such that 1, θ1, θ2 are Q-linearly independent.Consider two systems (Xθ1 ×T, gθ1 ×Rθ2) and (Xθ2 ×T, gθ2 ×Rθ1). Then both are minimal. LetA = C∗(Xθ1 × T, gθ1 × Rθ2) and B = C∗(Xθ2 × T, gθ2 × Rθ1). It follows from [Ph3, Proposition1.12] that

K1(A) ∼= K1(B) = Z3

and

(K0(A),K0(A)+, [1A]) ∼= (Z + θ1Z + θ2Z, (Z + θ1Z + θ2Z)+, 1) ∼= (K0(B), K0(B)+, [1B]),

where Z + θ1Z + θ2Z ⊂ R. In particular, both systems are rigid. Moreover A and B have tracialrank zero and they are isomorphic by the classification theorem in [L5]. However, there is noorder isomorphism between Z + θ1Z and Z + θ2Z. It follows from Theorem 7.9 that they arenot approximately K-conjugate. On the other hand, by [M3, Corollary 4.10], they are weaklyapproximately conjugate.

When we choose another continuous function ξ : Xθ1 → T, two systems gθ1×Rθ2 and gθ1×Rξ

may not be conjugate. But, if the integral value of ξ is equal to θ2, then we can conclude thatthey are approximately K-conjugate by Theorem 7.8.

Example 9.3. We would like to construct a non-orientation preserving minimal homeomor-phism on X × T concretely. Let (X, α) be an odometer system of type 3∞. It is well-knownthat K0(X, α) is isomorphic to Z[1/3]. We regard X as a projective limit of Z3n and denote

39

the canonical projection from X to Z3n by πn. Notice that πn(α(x)) = πn(x) + 1 for all x ∈ X,where the addition is understood modulo 3n. Let x0 be the point of X such that πn(x0) = 0 forall n ∈ N. We will construct a continuous map ϕ : X → Isom(T) of the form ϕx = λRξ(x) sothat α × ϕ is a minimal homeomorphism on X × T. Note that o(ϕ)(x) = 1 and [o(ϕ)] = 0 inK0(X, α)/2K0(X, α) ∼= Z2. By Lemma 8.5, if the closure of

{(α × ϕ)m(x0, 0) : m ∈ N}

contains {x0}×T, then we can deduce the minimality of α×ϕ. Let {tn}n∈N be a dense sequenceof T. Since α3n

(x0) → x0 as n → ∞, it suffices to construct ϕ so that (α × ϕ)3n(x0, 0) =

(α3n(x0),−tn).

Let sn ∈ (−2−1, 2−1] be the real number satisfying sn + Z = tn − tn−1, where we put t0 = 0.We define a map ξn : X → T by

ξn(x) =

{0 πn(x) = 0, 1, . . . , 3n−1 − 1(−1)k

2·3n−1 sn + Z otherwise.

For all n,m ∈ N, it is not hard to see that

3n−1∑

k=0

(−1)kξm(αk(x0)) =

{0 n < m

sm + Z n ≥ m.

Since |ξn(x)| < 3−n for every x ∈ X,

ξ(x) =∞∑

n=1

ξn(x)

exists and is continuous on X. Put ϕx = λRξ(x) for all x ∈ X. Then

3n−1∑

k=0

(−1)kξ(αk(x0)) =n∑

i=1

si + Z = tn

implies(α × ϕ)3

n(x0, 0) = (α3n

(x0), λ(tn)) = (α3n(x0),−tn)

for all n ∈ N. It follows that α × ϕ is minimal.

Example 9.4. We would like to construct a cocycle with values in Homeo+(T) which is notcohomologous to a cocycle with values in the rotation group. It is useful to introduce a completemetric d(·, ·) of Homeo+(T) defined by

d(ϕ,ψ) = maxt∈T

{|ϕ(t) − ψ(t)|, |ϕ−1(t) − ψ−1(t)|

}.

In the argument below, we use the following facts.

Fact (a). For any s, t ∈ T and ε > 0, there exists ρ ∈ Homeo+(T) such that |ρ(s) − s| < ε,|ρ(t) − s| < ε and ρ is conjugate to an irrational rotation.

Fact (b). Homeo+(T) is arcwise connected.

Let us construct two sequences of natural numbers mn and ln, and a sequence of maps ϕn :Zmn → Homeo+(T) inductively so that the following conditions are satisfied. It is convenient toview ϕn as a periodic map from Z.

40

(1) mn−1 divides mn and ln is not greater than mn/mn−1.

(2) ψn = ϕn(mn − 1) . . . ϕn(1)ϕn(0) is conjugate to an irrational rotation.

(3) Both |ψn(0)| and |ψn(1/2)| are less than 1/n.

(4) For every t ∈ T, {ψkn−1(t) : k = 1, 2, . . . , ln} is 1/n-dense in T.

(5) For every k = 0, 1, . . . , lnmn−1 − 1, ϕn(k) = ϕn−1(k).

(6) For every k ∈ Zmn , d(ϕn(k), ϕn−1(k)) is less than 2−n.

If these conditions are achieved, then we can finish the proof as follows. Let (X,α) be theodometer system of type {mn}n. Namely, X is the projective limit of Zmn and there existsa natural projection πn : X → Zmn such that πn(α(x)) = πn(x) + 1, where the addition isunderstood modulo mn. Let x0 be the point of X such that πn(x0) = 0 for all n ∈ N. From (6),

ϕx = limn→∞

ϕn(πn(x)) ∈ Homeo+(T)

exists for all x ∈ X and ϕ is a continuous map from X to Homeo+(T). By (4) and (5), for alln ∈ N and t ∈ T, we can see that

{ϕαk−1(x0) . . . ϕα(x0)ϕx0(t) : k = mn−1, 2mn−1, . . . , lnmn−1}

is 1/n-dense in T. Hence, for every t ∈ T, the closure of

{(α × ϕ)k(x0, t) : k ∈ N}

in X × T contains {x0} × T. It follows that α × ϕ is minimal. Let us check that ϕ is nevercohomologous to a cocycle with values in the rotation group. Suppose that ϕ is cohomologousto a cocycle with values in the rotation group. Then we would have a homeomorphism id×γ onX × T such that

(id×γ)(α × ϕ) = (α × ϕ)(id×γ)

and (id×γ)(x0, 0) = (x0, 1/2). By (3), we can verify that

(α × ϕ)mn(x0, 0) → (x0, 0)

and(α × ϕ)mn(x0, 1/2) → (x0, 0)

in X × T as n → ∞. Consequently we obtain

(id×γ)(x0, 0) = limn→∞

(id×γ)(α × ϕ)mn(x0, 0)

= limn→∞

(α × ϕ)mn(id×γ)(x0, 0)

= limn→∞

(α × ϕ)mn(x0, 1/2) = (x0, 0),

which is a contradiction.Let us construct mn, ln and ϕn. Put m1 = l1 = 1 and let ϕ1 be an irrational rotation.

Suppose that mn−1, kn−1 and ϕn−1 have been fixed. Put

ψn−1 = ϕn−1(mn−1 − 1) . . . ϕn−1(1)ϕn−1(0).

By (2), there exists ω ∈ Homeo+(T) such that ωψn−1ω−1 = Rθ, where r(ψn−1) = θ is an

irrational number. Since ψn−1 is minimal, we can find a natural number ln so that the condition

41

(4) holds. Applying Fact (a) to ω(0) and ω(1/2), we obtain ρ ∈ Homeo+(T) such that both|ω−1ρω(0)| and |ω−1ρω(1/2)| are less than 1/n. Furthermore ρ is conjugate to an irrationalrotation. Define ω : Zmn−1 → Homeo+(T) by

ω(k) = ωϕn−1(0)−1ϕn−1(1)−1 . . . ϕn−1(k − 1)−1

for all k = 0, 1, . . . ,mn−1 − 1. Evidently we have

ω(k + 1)ϕn−1(k)ω(k)−1 =

{id k = mn−1 − 1Rθ k = mn−1 − 1

for all k ∈ Zmn−1 . Choose ε > 0 so that d(ϕ, id) < ε implies

d(ϕn−1(k)ω(k)−1ϕω(k), ϕn−1(k)) <12n

for all k ∈ Zmn−1 . By Fact (b), we can find a natural number N greater than ln and a sequenceof homeomorphisms

id = ρ0, ρ1, . . . , ρmn−1(N−ln) = ρR−Nθ

such thatd(ρi+1ρ

−1i , id) < ε

for all i = 0, 1, . . . ,mn−1(N − ln) − 1. Note that this is easily done because d(·, ·) is invariantunder rotations. Put mn = mn−1N . We define a map ρ : Zmn → Homeo+(T) by

ρ(k) =

{id k = 0, 1, . . . , lnmn−1 − 1R−j

θ ρk′+1ρ−1k′ Rj

θ otherwise,

where k′ = k− lnmn−1 and j is a natural number satisfying N −k/mn−1 ≤ j < N +1−k/mn−1.Notice that we still have

d(ρ(k), id) < ε

for all k ∈ Zmn . Define ϕn : Zmn → Homeo+(T) by

ϕn(k) = ϕn−1(k)ω(k)−1ρ(k)ω(k).

The condition (5) is already built in this definition. The condition (6) is immediate from thechoice of ε. Since one can check that

ψn = ϕn(mn − 1) . . . ϕn(1)ϕn(0)

= ω(mn)−1 (Rθρ(mn − 1) . . . ρ(mn − mn−1)Rθ . . . ρ(lnmn−1))Rlnθ ω(0)

= ω−1ρR−Nθ RN

θ ω

= ω−1ρω,

the conditions (2) and (3) follow immediately.

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Huaxin Line-mail: hxlin@noether.uoregon.eduDepartment of MathematicsUniversity of OregonEugene, Oregon 97403U.S.A.

Hiroki Matuie-mail: matui@math.s.chiba-u.ac.jpGraduate School of Science and Technology,Chiba University,1-33 Yayoi-cho, Inage-ku,Chiba 263-8522,Japan.

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